:: NDIFF_1 semantic presentation

theorem Th1: :: NDIFF_1:1
for b1 being RealNormSpace
for b2 being Point of b1
for b3, b4 being Neighbourhood of b2 holds
ex b5 being Neighbourhood of b2 st
( b5 c= b3 & b5 c= b4 )
proof end;

theorem Th2: :: NDIFF_1:2
for b1 being RealNormSpace
for b2 being Subset of b1 holds
( b2 is open implies for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being Neighbourhood of b3 holds
not b4 c= b2 ) ) )
proof end;

theorem Th3: :: NDIFF_1:3
for b1 being RealNormSpace
for b2 being Subset of b1 holds
( b2 is open implies for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being Real holds
not ( 0 < b4 & { b5 where B is Point of b1 : ||.(b5 - b3).|| < b4 } c= b2 ) ) ) )
proof end;

theorem Th4: :: NDIFF_1:4
for b1 being RealNormSpace
for b2 being Subset of b1 holds
( ( for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being Neighbourhood of b3 holds
not b4 c= b2 ) ) ) implies b2 is open )
proof end;

theorem Th5: :: NDIFF_1:5
for b1 being RealNormSpace
for b2 being Subset of b1 holds
( ( for b3 being Point of b1 holds
not ( b3 in b2 & ( for b4 being Neighbourhood of b3 holds
not b4 c= b2 ) ) ) iff b2 is open ) by Th2, Th4;

definition
let c1 be ZeroStr ;
let c2 be sequence of c1;
attr a2 is being_not_0 means :Def1: :: NDIFF_1:def 1
rng a2 c= the carrier of a1 \ {(0. a1)};
end;

:: deftheorem Def1 defines being_not_0 NDIFF_1:def 1 :
for b1 being ZeroStr
for b2 being sequence of b1 holds
( b2 is being_not_0 iff rng b2 c= the carrier of b1 \ {(0. b1)} );

notation
let c1 be ZeroStr ;
let c2 be sequence of c1;
synonym c2 is_not_0 for being_not_0 c2;
end;

theorem Th6: :: NDIFF_1:6
for b1 being RealNormSpace
for b2 being sequence of b1 holds
( b2 is being_not_0 iff for b3 being set holds
not ( b3 in NAT & not b2 . b3 <> 0. b1 ) )
proof end;

theorem Th7: :: NDIFF_1:7
for b1 being RealNormSpace
for b2 being sequence of b1 holds
( b2 is being_not_0 iff for b3 being Nat holds
b2 . b3 <> 0. b1 )
proof end;

definition
let c1 be RealLinearSpace;
let c2 be sequence of c1;
let c3 be Real_Sequence;
func c3 (#) c2 -> sequence of a1 means :Def2: :: NDIFF_1:def 2
for b1 being Nat holds a4 . b1 = (a3 . b1) * (a2 . b1);
existence
ex b1 being sequence of c1 st
for b2 being Nat holds b1 . b2 = (c3 . b2) * (c2 . b2)
proof end;
uniqueness
for b1, b2 being sequence of c1 holds
( ( for b3 being Nat holds b1 . b3 = (c3 . b3) * (c2 . b3) ) & ( for b3 being Nat holds b2 . b3 = (c3 . b3) * (c2 . b3) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def2 defines (#) NDIFF_1:def 2 :
for b1 being RealLinearSpace
for b2 being sequence of b1
for b3 being Real_Sequence
for b4 being sequence of b1 holds
( b4 = b3 (#) b2 iff for b5 being Nat holds b4 . b5 = (b3 . b5) * (b2 . b5) );

definition
let c1 be RealLinearSpace;
let c2 be Point of c1;
let c3 be Real_Sequence;
func c3 * c2 -> sequence of a1 means :Def3: :: NDIFF_1:def 3
for b1 being Nat holds a4 . b1 = (a3 . b1) * a2;
existence
ex b1 being sequence of c1 st
for b2 being Nat holds b1 . b2 = (c3 . b2) * c2
proof end;
uniqueness
for b1, b2 being sequence of c1 holds
( ( for b3 being Nat holds b1 . b3 = (c3 . b3) * c2 ) & ( for b3 being Nat holds b2 . b3 = (c3 . b3) * c2 ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines * NDIFF_1:def 3 :
for b1 being RealLinearSpace
for b2 being Point of b1
for b3 being Real_Sequence
for b4 being sequence of b1 holds
( b4 = b3 * b2 iff for b5 being Nat holds b4 . b5 = (b3 . b5) * b2 );

theorem Th8: :: NDIFF_1:8
for b1 being RealNormSpace
for b2 being sequence of b1
for b3, b4 being Real_Sequence holds (b3 + b4) (#) b2 = (b3 (#) b2) + (b4 (#) b2)
proof end;

theorem Th9: :: NDIFF_1:9
for b1 being RealNormSpace
for b2 being Real_Sequence
for b3, b4 being sequence of b1 holds b2 (#) (b3 + b4) = (b2 (#) b3) + (b2 (#) b4)
proof end;

theorem Th10: :: NDIFF_1:10
for b1 being Real
for b2 being RealNormSpace
for b3 being sequence of b2
for b4 being Real_Sequence holds b1 * (b4 (#) b3) = b4 (#) (b1 * b3)
proof end;

theorem Th11: :: NDIFF_1:11
for b1 being RealNormSpace
for b2 being sequence of b1
for b3, b4 being Real_Sequence holds (b3 - b4) (#) b2 = (b3 (#) b2) - (b4 (#) b2)
proof end;

theorem Th12: :: NDIFF_1:12
for b1 being RealNormSpace
for b2 being Real_Sequence
for b3, b4 being sequence of b1 holds b2 (#) (b3 - b4) = (b2 (#) b3) - (b2 (#) b4)
proof end;

theorem Th13: :: NDIFF_1:13
for b1 being RealNormSpace
for b2 being Real_Sequence
for b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies b2 (#) b3 is convergent )
proof end;

theorem Th14: :: NDIFF_1:14
for b1 being RealNormSpace
for b2 being Real_Sequence
for b3 being sequence of b1 holds
( b2 is convergent & b3 is convergent implies lim (b2 (#) b3) = (lim b2) * (lim b3) )
proof end;

theorem Th15: :: NDIFF_1:15
for b1 being Nat
for b2 being RealNormSpace
for b3, b4 being sequence of b2 holds (b3 + b4) ^\ b1 = (b3 ^\ b1) + (b4 ^\ b1)
proof end;

theorem Th16: :: NDIFF_1:16
for b1 being Nat
for b2 being RealNormSpace
for b3, b4 being sequence of b2 holds (b3 - b4) ^\ b1 = (b3 ^\ b1) - (b4 ^\ b1)
proof end;

theorem Th17: :: NDIFF_1:17
for b1 being Nat
for b2 being RealNormSpace
for b3 being sequence of b2 holds
( b3 is_not_0 implies b3 ^\ b1 is_not_0 )
proof end;

theorem Th18: :: NDIFF_1:18
for b1 being Nat
for b2 being RealNormSpace
for b3 being sequence of b2 holds
b3 ^\ b1 is subsequence of b3
proof end;

theorem Th19: :: NDIFF_1:19
for b1 being RealNormSpace
for b2, b3 being sequence of b1 holds
( b2 is constant & b3 is subsequence of b2 implies b3 is constant )
proof end;

theorem Th20: :: NDIFF_1:20
for b1 being RealNormSpace
for b2, b3 being sequence of b1 holds
( b2 is constant & b3 is subsequence of b2 implies b2 = b3 )
proof end;

definition
let c1 be RealNormSpace;
let c2 be sequence of c1;
attr a2 is convergent_to_0 means :Def4: :: NDIFF_1:def 4
( a2 is_not_0 & a2 is convergent & lim a2 = 0. a1 );
end;

:: deftheorem Def4 defines convergent_to_0 NDIFF_1:def 4 :
for b1 being RealNormSpace
for b2 being sequence of b1 holds
( b2 is convergent_to_0 iff ( b2 is_not_0 & b2 is convergent & lim b2 = 0. b1 ) );

theorem Th21: :: NDIFF_1:21
for b1 being RealNormSpace
for b2 being sequence of b1 holds
( b2 is constant implies ( b2 is convergent & ( for b3 being Nat holds lim b2 = b2 . b3 ) ) )
proof end;

theorem Th22: :: NDIFF_1:22
for b1 being RealNormSpace
for b2 being sequence of b1
for b3 being Point of b1
for b4 being Real holds
( 0 < b4 & ( for b5 being Nat holds b2 . b5 = (1 / (b5 + b4)) * b3 ) implies b2 is convergent )
proof end;

theorem Th23: :: NDIFF_1:23
for b1 being RealNormSpace
for b2 being sequence of b1
for b3 being Point of b1
for b4 being Real holds
( 0 < b4 & ( for b5 being Nat holds b2 . b5 = (1 / (b5 + b4)) * b3 ) implies lim b2 = 0. b1 )
proof end;

theorem Th24: :: NDIFF_1:24
for b1 being RealNormSpace
for b2 being convergent_to_0 Real_Sequence
for b3 being Point of b1 holds
( b3 <> 0. b1 implies b2 * b3 is convergent_to_0 )
proof end;

theorem Th25: :: NDIFF_1:25
for b1 being RealNormSpace
for b2 being Subset of b1 holds
( ( for b3 being Point of b1 holds
( b3 in b2 iff b3 in the carrier of b1 ) ) iff b2 = the carrier of b1 )
proof end;

registration
let c1 be non trivial RealNormSpace;
cluster convergent_to_0 Relation of NAT ,the carrier of a1;
existence
ex b1 being sequence of c1 st b1 is convergent_to_0
proof end;
end;

registration
let c1 be non trivial RealNormSpace;
cluster V6 Relation of NAT ,the carrier of a1;
existence
ex b1 being sequence of c1 st b1 is constant
proof end;
end;

definition
let c1, c2 be non trivial RealNormSpace;
let c3 be PartFunc of c1,c2;
attr a3 is REST-like means :Def5: :: NDIFF_1:def 5
( a3 is total & ( for b1 being convergent_to_0 sequence of a1 holds
( (||.b1.|| " ) (#) (a3 * b1) is convergent & lim ((||.b1.|| " ) (#) (a3 * b1)) = 0. a2 ) ) );
end;

:: deftheorem Def5 defines REST-like NDIFF_1:def 5 :
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2 holds
( b3 is REST-like iff ( b3 is total & ( for b4 being convergent_to_0 sequence of b1 holds
( (||.b4.|| " ) (#) (b3 * b4) is convergent & lim ((||.b4.|| " ) (#) (b3 * b4)) = 0. b2 ) ) ) );

registration
let c1, c2 be non trivial RealNormSpace;
cluster REST-like Relation of the carrier of a1,the carrier of a2;
existence
ex b1 being PartFunc of c1,c2 st b1 is REST-like
proof end;
end;

definition
let c1, c2 be non trivial RealNormSpace;
mode REST is REST-like PartFunc of a1,a2;
end;

theorem Th26: :: NDIFF_1:26
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2 holds
( b3 is total implies ( b3 is REST-like iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being Real holds
not ( b5 > 0 & ( for b6 being Point of b1 holds
not ( b6 <> 0. b1 & ||.b6.|| < b5 & not (||.b6.|| " ) * ||.(b3 /. b6).|| < b4 ) ) ) ) ) ) )
proof end;

theorem Th27: :: NDIFF_1:27
for b1, b2 being non trivial RealNormSpace
for b3 being REST of b1,b2
for b4 being convergent_to_0 sequence of b1 holds
( b3 * b4 is convergent & lim (b3 * b4) = 0. b2 )
proof end;

theorem Th28: :: NDIFF_1:28
for b1 being Nat
for b2 being non trivial RealNormSpace
for b3 being sequence of b2 holds rng (b3 ^\ b1) c= rng b3
proof end;

theorem Th29: :: NDIFF_1:29
for b1 being Nat
for b2, b3 being non trivial RealNormSpace
for b4 being PartFunc of b2,b3
for b5 being sequence of b2 holds
( rng b5 c= dom b4 implies (b4 * b5) ^\ b1 = b4 * (b5 ^\ b1) )
proof end;

theorem Th30: :: NDIFF_1:30
for b1, b2 being non trivial RealNormSpace
for b3, b4 being PartFunc of b1,b2
for b5 being sequence of b1 holds
( b3 is total & b4 is total implies ( (b3 + b4) * b5 = (b3 * b5) + (b4 * b5) & (b3 - b4) * b5 = (b3 * b5) - (b4 * b5) ) )
proof end;

theorem Th31: :: NDIFF_1:31
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being sequence of b1
for b5 being Real holds
( b3 is total implies (b5 (#) b3) * b4 = b5 * (b3 * b4) )
proof end;

theorem Th32: :: NDIFF_1:32
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b2,b1
for b4 being Point of b2 holds
( b3 is_continuous_in b4 iff ( b4 in dom b3 & ( for b5 being sequence of b2 holds
( rng b5 c= dom b3 & b5 is convergent & lim b5 = b4 & ( for b6 being Nat holds
b5 . b6 <> b4 ) implies ( b3 * b5 is convergent & b3 /. b4 = lim (b3 * b5) ) ) ) ) )
proof end;

theorem Th33: :: NDIFF_1:33
for b1, b2 being non trivial RealNormSpace
for b3, b4 being REST of b2,b1 holds
( b3 + b4 is REST of b2,b1 & b3 - b4 is REST of b2,b1 )
proof end;

theorem Th34: :: NDIFF_1:34
for b1, b2 being non trivial RealNormSpace
for b3 being Real
for b4 being REST of b2,b1 holds
b3 (#) b4 is REST of b2,b1
proof end;

definition
let c1, c2 be non trivial RealNormSpace;
let c3 be PartFunc of c1,c2;
let c4 be Point of c1;
pred c3 is_differentiable_in c4 means :Def6: :: NDIFF_1:def 6
ex b1 being Neighbourhood of a4 st
( b1 c= dom a3 & ex b2 being Point of (R_NormSpace_of_BoundedLinearOperators a1,a2)ex b3 being REST of a1,a2 st
for b4 being Point of a1 holds
( b4 in b1 implies (a3 /. b4) - (a3 /. a4) = (b2 . (b4 - a4)) + (b3 /. (b4 - a4)) ) );
end;

:: deftheorem Def6 defines is_differentiable_in NDIFF_1:def 6 :
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Point of b1 holds
( b3 is_differentiable_in b4 iff ex b5 being Neighbourhood of b4 st
( b5 c= dom b3 & ex b6 being Point of (R_NormSpace_of_BoundedLinearOperators b1,b2)ex b7 being REST of b1,b2 st
for b8 being Point of b1 holds
( b8 in b5 implies (b3 /. b8) - (b3 /. b4) = (b6 . (b8 - b4)) + (b7 /. (b8 - b4)) ) ) );

definition
let c1, c2 be non trivial RealNormSpace;
let c3 be PartFunc of c1,c2;
let c4 be Point of c1;
assume E34: c3 is_differentiable_in c4 ;
func diff c3,c4 -> Point of (R_NormSpace_of_BoundedLinearOperators a1,a2) means :Def7: :: NDIFF_1:def 7
ex b1 being Neighbourhood of a4 st
( b1 c= dom a3 & ex b2 being REST of a1,a2 st
for b3 being Point of a1 holds
( b3 in b1 implies (a3 /. b3) - (a3 /. a4) = (a5 . (b3 - a4)) + (b2 /. (b3 - a4)) ) );
existence
ex b1 being Point of (R_NormSpace_of_BoundedLinearOperators c1,c2)ex b2 being Neighbourhood of c4 st
( b2 c= dom c3 & ex b3 being REST of c1,c2 st
for b4 being Point of c1 holds
( b4 in b2 implies (c3 /. b4) - (c3 /. c4) = (b1 . (b4 - c4)) + (b3 /. (b4 - c4)) ) )
proof end;
uniqueness
for b1, b2 being Point of (R_NormSpace_of_BoundedLinearOperators c1,c2) holds
( ex b3 being Neighbourhood of c4 st
( b3 c= dom c3 & ex b4 being REST of c1,c2 st
for b5 being Point of c1 holds
( b5 in b3 implies (c3 /. b5) - (c3 /. c4) = (b1 . (b5 - c4)) + (b4 /. (b5 - c4)) ) ) & ex b3 being Neighbourhood of c4 st
( b3 c= dom c3 & ex b4 being REST of c1,c2 st
for b5 being Point of c1 holds
( b5 in b3 implies (c3 /. b5) - (c3 /. c4) = (b2 . (b5 - c4)) + (b4 /. (b5 - c4)) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def7 defines diff NDIFF_1:def 7 :
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Point of b1 holds
( b3 is_differentiable_in b4 implies for b5 being Point of (R_NormSpace_of_BoundedLinearOperators b1,b2) holds
( b5 = diff b3,b4 iff ex b6 being Neighbourhood of b4 st
( b6 c= dom b3 & ex b7 being REST of b1,b2 st
for b8 being Point of b1 holds
( b8 in b6 implies (b3 /. b8) - (b3 /. b4) = (b5 . (b8 - b4)) + (b7 /. (b8 - b4)) ) ) ) );

definition
let c1 be set ;
let c2, c3 be non trivial RealNormSpace;
let c4 be PartFunc of c2,c3;
pred c4 is_differentiable_on c1 means :Def8: :: NDIFF_1:def 8
( a1 c= dom a4 & ( for b1 being Point of a2 holds
( b1 in a1 implies a4 | a1 is_differentiable_in b1 ) ) );
end;

:: deftheorem Def8 defines is_differentiable_on NDIFF_1:def 8 :
for b1 being set
for b2, b3 being non trivial RealNormSpace
for b4 being PartFunc of b2,b3 holds
( b4 is_differentiable_on b1 iff ( b1 c= dom b4 & ( for b5 being Point of b2 holds
( b5 in b1 implies b4 | b1 is_differentiable_in b5 ) ) ) );

theorem Th35: :: NDIFF_1:35
for b1 being set
for b2, b3 being non trivial RealNormSpace
for b4 being PartFunc of b2,b3 holds
( b4 is_differentiable_on b1 implies b1 is Subset of b2 )
proof end;

theorem Th36: :: NDIFF_1:36
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Subset of b1 holds
( b4 is open implies ( b3 is_differentiable_on b4 iff ( b4 c= dom b3 & ( for b5 being Point of b1 holds
( b5 in b4 implies b3 is_differentiable_in b5 ) ) ) ) )
proof end;

theorem Th37: :: NDIFF_1:37
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Subset of b1 holds
( b3 is_differentiable_on b4 implies b4 is open )
proof end;

definition
let c1, c2 be non trivial RealNormSpace;
let c3 be PartFunc of c1,c2;
let c4 be set ;
assume E38: c3 is_differentiable_on c4 ;
func c3 `| c4 -> PartFunc of a1,(R_NormSpace_of_BoundedLinearOperators a1,a2) means :Def9: :: NDIFF_1:def 9
( dom a5 = a4 & ( for b1 being Point of a1 holds
( b1 in a4 implies a5 /. b1 = diff a3,b1 ) ) );
existence
ex b1 being PartFunc of c1,(R_NormSpace_of_BoundedLinearOperators c1,c2) st
( dom b1 = c4 & ( for b2 being Point of c1 holds
( b2 in c4 implies b1 /. b2 = diff c3,b2 ) ) )
proof end;
uniqueness
for b1, b2 being PartFunc of c1,(R_NormSpace_of_BoundedLinearOperators c1,c2) holds
( dom b1 = c4 & ( for b3 being Point of c1 holds
( b3 in c4 implies b1 /. b3 = diff c3,b3 ) ) & dom b2 = c4 & ( for b3 being Point of c1 holds
( b3 in c4 implies b2 /. b3 = diff c3,b3 ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def9 defines `| NDIFF_1:def 9 :
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being set holds
( b3 is_differentiable_on b4 implies for b5 being PartFunc of b1,(R_NormSpace_of_BoundedLinearOperators b1,b2) holds
( b5 = b3 `| b4 iff ( dom b5 = b4 & ( for b6 being Point of b1 holds
( b6 in b4 implies b5 /. b6 = diff b3,b6 ) ) ) ) );

theorem Th38: :: NDIFF_1:38
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Subset of b1 holds
( b4 is open & b4 c= dom b3 & ex b5 being Point of b2 st rng b3 = {b5} implies ( b3 is_differentiable_on b4 & ( for b5 being Point of b1 holds
( b5 in b4 implies (b3 `| b4) /. b5 = 0. (R_NormSpace_of_BoundedLinearOperators b1,b2) ) ) ) )
proof end;

registration
let c1 be non trivial RealNormSpace;
let c2 be convergent_to_0 sequence of c1;
let c3 be Nat;
cluster a2 ^\ a3 -> convergent_to_0 ;
coherence
c2 ^\ c3 is convergent_to_0
proof end;
end;

registration
let c1 be non trivial RealNormSpace;
let c2 be V6 sequence of c1;
let c3 be Nat;
cluster a2 ^\ a3 -> V6 ;
coherence
c2 ^\ c3 is constant
proof end;
end;

theorem Th39: :: NDIFF_1:39
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Point of b1
for b5 being Neighbourhood of b4 holds
( b3 is_differentiable_in b4 & b5 c= dom b3 implies for b6 being convergent_to_0 sequence of b1
for b7 being V6 sequence of b1 holds
( rng b7 = {b4} & rng (b6 + b7) c= b5 implies ( (b3 * (b6 + b7)) - (b3 * b7) is convergent & lim ((b3 * (b6 + b7)) - (b3 * b7)) = 0. b2 ) ) )
proof end;

theorem Th40: :: NDIFF_1:40
for b1, b2 being non trivial RealNormSpace
for b3, b4 being PartFunc of b2,b1
for b5 being Point of b2 holds
( b3 is_differentiable_in b5 & b4 is_differentiable_in b5 implies ( b3 + b4 is_differentiable_in b5 & diff (b3 + b4),b5 = (diff b3,b5) + (diff b4,b5) ) )
proof end;

theorem Th41: :: NDIFF_1:41
for b1, b2 being non trivial RealNormSpace
for b3, b4 being PartFunc of b2,b1
for b5 being Point of b2 holds
( b3 is_differentiable_in b5 & b4 is_differentiable_in b5 implies ( b3 - b4 is_differentiable_in b5 & diff (b3 - b4),b5 = (diff b3,b5) - (diff b4,b5) ) )
proof end;

theorem Th42: :: NDIFF_1:42
for b1, b2 being non trivial RealNormSpace
for b3 being Real
for b4 being PartFunc of b2,b1
for b5 being Point of b2 holds
( b4 is_differentiable_in b5 implies ( b3 (#) b4 is_differentiable_in b5 & diff (b3 (#) b4),b5 = b3 * (diff b4,b5) ) )
proof end;

theorem Th43: :: NDIFF_1:43
for b1 being non trivial RealNormSpace
for b2 being PartFunc of b1,b1
for b3 being Subset of b1 holds
( b3 is open & b3 c= dom b2 & b2 | b3 = id b3 implies ( b2 is_differentiable_on b3 & ( for b4 being Point of b1 holds
( b4 in b3 implies (b2 `| b3) /. b4 = id the carrier of b1 ) ) ) )
proof end;

theorem Th44: :: NDIFF_1:44
for b1, b2 being non trivial RealNormSpace
for b3 being Subset of b1 holds
( b3 is open implies for b4, b5 being PartFunc of b1,b2 holds
( b3 c= dom (b4 + b5) & b4 is_differentiable_on b3 & b5 is_differentiable_on b3 implies ( b4 + b5 is_differentiable_on b3 & ( for b6 being Point of b1 holds
( b6 in b3 implies ((b4 + b5) `| b3) /. b6 = (diff b4,b6) + (diff b5,b6) ) ) ) ) )
proof end;

theorem Th45: :: NDIFF_1:45
for b1, b2 being non trivial RealNormSpace
for b3 being Subset of b1 holds
( b3 is open implies for b4, b5 being PartFunc of b1,b2 holds
( b3 c= dom (b4 - b5) & b4 is_differentiable_on b3 & b5 is_differentiable_on b3 implies ( b4 - b5 is_differentiable_on b3 & ( for b6 being Point of b1 holds
( b6 in b3 implies ((b4 - b5) `| b3) /. b6 = (diff b4,b6) - (diff b5,b6) ) ) ) ) )
proof end;

theorem Th46: :: NDIFF_1:46
for b1, b2 being non trivial RealNormSpace
for b3 being Subset of b1 holds
( b3 is open implies for b4 being Real
for b5 being PartFunc of b1,b2 holds
( b3 c= dom (b4 (#) b5) & b5 is_differentiable_on b3 implies ( b4 (#) b5 is_differentiable_on b3 & ( for b6 being Point of b1 holds
( b6 in b3 implies ((b4 (#) b5) `| b3) /. b6 = b4 * (diff b5,b6) ) ) ) ) )
proof end;

theorem Th47: :: NDIFF_1:47
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b1,b2
for b4 being Subset of b1 holds
( b4 is open & b4 c= dom b3 & b3 is_constant_on b4 implies ( b3 is_differentiable_on b4 & ( for b5 being Point of b1 holds
( b5 in b4 implies (b3 `| b4) /. b5 = 0. (R_NormSpace_of_BoundedLinearOperators b1,b2) ) ) ) )
proof end;

theorem Th48: :: NDIFF_1:48
for b1 being non trivial RealNormSpace
for b2 being PartFunc of b1,b1
for b3 being Real
for b4 being Point of b1
for b5 being Subset of b1 holds
( b5 is open & b5 c= dom b2 & ( for b6 being Point of b1 holds
( b6 in b5 implies b2 /. b6 = (b3 * b6) + b4 ) ) implies ( b2 is_differentiable_on b5 & ( for b6 being Point of b1 holds
( b6 in b5 implies (b2 `| b5) /. b6 = b3 * (FuncUnit b1) ) ) ) )
proof end;

theorem Th49: :: NDIFF_1:49
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b2,b1
for b4 being Point of b2 holds
( b3 is_differentiable_in b4 implies b3 is_continuous_in b4 )
proof end;

theorem Th50: :: NDIFF_1:50
for b1 being set
for b2, b3 being non trivial RealNormSpace
for b4 being PartFunc of b2,b3 holds
( b4 is_differentiable_on b1 implies b4 is_continuous_on b1 )
proof end;

theorem Th51: :: NDIFF_1:51
for b1 being set
for b2, b3 being non trivial RealNormSpace
for b4 being PartFunc of b3,b2
for b5 being Subset of b3 holds
( b5 is open & b4 is_differentiable_on b1 & b5 c= b1 implies b4 is_differentiable_on b5 )
proof end;

theorem Th52: :: NDIFF_1:52
for b1, b2 being non trivial RealNormSpace
for b3 being PartFunc of b2,b1
for b4 being Point of b2 holds
not ( b3 is_differentiable_in b4 & ( for b5 being Neighbourhood of b4 holds
not ( b5 c= dom b3 & ex b6 being REST of b2,b1 st
( b6 /. (0. b2) = 0. b1 & b6 is_continuous_in 0. b2 & ( for b7 being Point of b2 holds
( b7 in b5 implies (b3 /. b7) - (b3 /. b4) = ((diff b3,b4) . (b7 - b4)) + (b6 /. (b7 - b4)) ) ) ) ) ) )
proof end;