:: BHSP_3 semantic presentation

deffunc H1( RealUnitarySpace) -> M2(the carrier of a1) = 0. a1;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
attr a2 is Cauchy means :Def1: :: BHSP_3:def 1
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
ex b3, b4 being Nat st
( b3 >= b2 & b4 >= b2 & not dist (a2 . b3),(a2 . b4) < b1 ) ) );
end;

:: deftheorem Def1 defines Cauchy BHSP_3:def 1 :
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is Cauchy iff for b3 being Real holds
not ( b3 > 0 & ( for b4 being Nat holds
ex b5, b6 being Nat st
( b5 >= b4 & b6 >= b4 & not dist (b2 . b5),(b2 . b6) < b3 ) ) ) );

notation
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
synonym c2 is_Cauchy_sequence for Cauchy c2;
end;

theorem Th1: :: BHSP_3:1
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is constant implies b2 is_Cauchy_sequence )
proof end;

theorem Th2: :: BHSP_3:2
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is_Cauchy_sequence iff for b3 being Real holds
not ( b3 > 0 & ( for b4 being Nat holds
ex b5, b6 being Nat st
( b5 >= b4 & b6 >= b4 & not ||.((b2 . b5) - (b2 . b6)).|| < b3 ) ) ) )
proof end;

theorem Th3: :: BHSP_3:3
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_Cauchy_sequence & b3 is_Cauchy_sequence implies b2 + b3 is_Cauchy_sequence )
proof end;

theorem Th4: :: BHSP_3:4
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_Cauchy_sequence & b3 is_Cauchy_sequence implies b2 - b3 is_Cauchy_sequence )
proof end;

theorem Th5: :: BHSP_3:5
for b1 being RealUnitarySpace
for b2 being Real
for b3 being sequence of b1 holds
( b3 is_Cauchy_sequence implies b2 * b3 is_Cauchy_sequence )
proof end;

theorem Th6: :: BHSP_3:6
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is_Cauchy_sequence implies - b2 is_Cauchy_sequence )
proof end;

theorem Th7: :: BHSP_3:7
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is_Cauchy_sequence implies b3 + b2 is_Cauchy_sequence )
proof end;

theorem Th8: :: BHSP_3:8
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3 being sequence of b1 holds
( b3 is_Cauchy_sequence implies b3 - b2 is_Cauchy_sequence )
proof end;

theorem Th9: :: BHSP_3:9
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies b2 is_Cauchy_sequence )
proof end;

definition
let c1 be RealUnitarySpace;
let c2, c3 be sequence of c1;
pred c2 is_compared_to c3 means :Def2: :: BHSP_3:def 2
for b1 being Real holds
not ( b1 > 0 & ( for b2 being Nat holds
ex b3 being Nat st
( b3 >= b2 & not dist (a2 . b3),(a3 . b3) < b1 ) ) );
end;

:: deftheorem Def2 defines is_compared_to BHSP_3:def 2 :
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_compared_to b3 iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not dist (b2 . b6),(b3 . b6) < b4 ) ) ) );

theorem Th10: :: BHSP_3:10
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds b2 is_compared_to b2
proof end;

theorem Th11: :: BHSP_3:11
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_compared_to b3 implies b3 is_compared_to b2 )
proof end;

definition
let c1 be RealUnitarySpace;
let c2, c3 be sequence of c1;
redefine pred is_compared_to as c2 is_compared_to c3;
reflexivity
for b1 being sequence of c1 holds b1 is_compared_to b1
by Th10;
symmetry
for b1, b2 being sequence of c1 holds
( b1 is_compared_to b2 implies b2 is_compared_to b1 )
by Th11;
end;

theorem Th12: :: BHSP_3:12
for b1 being RealUnitarySpace
for b2, b3, b4 being sequence of b1 holds
( b2 is_compared_to b3 & b3 is_compared_to b4 implies b2 is_compared_to b4 )
proof end;

theorem Th13: :: BHSP_3:13
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_compared_to b3 iff for b4 being Real holds
not ( b4 > 0 & ( for b5 being Nat holds
ex b6 being Nat st
( b6 >= b5 & not ||.((b2 . b6) - (b3 . b6)).|| < b4 ) ) ) )
proof end;

theorem Th14: :: BHSP_3:14
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( ex b4 being Nat st
for b5 being Nat holds
( b5 >= b4 implies b2 . b5 = b3 . b5 ) implies b2 is_compared_to b3 )
proof end;

theorem Th15: :: BHSP_3:15
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_Cauchy_sequence & b2 is_compared_to b3 implies b3 is_Cauchy_sequence )
proof end;

theorem Th16: :: BHSP_3:16
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b2 is_compared_to b3 implies b3 is convergent )
proof end;

theorem Th17: :: BHSP_3:17
for b1 being RealUnitarySpace
for b2 being Point of b1
for b3, b4 being sequence of b1 holds
( b3 is convergent & lim b3 = b2 & b3 is_compared_to b4 implies ( b4 is convergent & lim b4 = b2 ) )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
attr a2 is bounded means :Def3: :: BHSP_3:def 3
ex b1 being Real st
( b1 > 0 & ( for b2 being Nat holds ||.(a2 . b2).|| <= b1 ) );
end;

:: deftheorem Def3 defines bounded BHSP_3:def 3 :
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is bounded iff ex b3 being Real st
( b3 > 0 & ( for b4 being Nat holds ||.(b2 . b4).|| <= b3 ) ) );

theorem Th18: :: BHSP_3:18
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is bounded & b3 is bounded implies b2 + b3 is bounded )
proof end;

theorem Th19: :: BHSP_3:19
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is bounded implies - b2 is bounded )
proof end;

theorem Th20: :: BHSP_3:20
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is bounded & b3 is bounded implies b2 - b3 is bounded )
proof end;

theorem Th21: :: BHSP_3:21
for b1 being RealUnitarySpace
for b2 being Real
for b3 being sequence of b1 holds
( b3 is bounded implies b2 * b3 is bounded )
proof end;

theorem Th22: :: BHSP_3:22
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is constant implies b2 is bounded )
proof end;

theorem Th23: :: BHSP_3:23
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
ex b4 being Real st
( b4 > 0 & ( for b5 being Nat holds
not ( b5 <= b3 & not ||.(b2 . b5).|| < b4 ) ) )
proof end;

theorem Th24: :: BHSP_3:24
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b2 is convergent implies b2 is bounded )
proof end;

theorem Th25: :: BHSP_3:25
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is bounded & b2 is_compared_to b3 implies b3 is bounded )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be increasing Seq_of_Nat;
let c3 be sequence of c1;
redefine func * as c3 * c2 -> sequence of a1;
coherence
c2 * c3 is sequence of c1
proof end;
end;

definition
let c1 be 1-sorted ;
let c2 be sequence of c1;
mode subsequence of c2 -> sequence of a1 means :Def4: :: BHSP_3:def 4
ex b1 being increasing Seq_of_Nat st a3 = a2 * b1;
existence
ex b1 being sequence of c1ex b2 being increasing Seq_of_Nat st b1 = c2 * b2
proof end;
end;

:: deftheorem Def4 defines subsequence BHSP_3:def 4 :
for b1 being 1-sorted
for b2, b3 being sequence of b1 holds
( b3 is subsequence of b2 iff ex b4 being increasing Seq_of_Nat st b3 = b2 * b4 );

theorem Th26: :: BHSP_3:26
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being increasing Seq_of_Nat
for b4 being Nat holds (b2 * b3) . b4 = b2 . (b3 . b4)
proof end;

theorem Th27: :: BHSP_3:27
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
b2 is subsequence of b2
proof end;

theorem Th28: :: BHSP_3:28
for b1 being RealUnitarySpace
for b2, b3, b4 being sequence of b1 holds
( b2 is subsequence of b3 & b3 is subsequence of b4 implies b2 is subsequence of b4 )
proof end;

theorem Th29: :: BHSP_3:29
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is constant & b3 is subsequence of b2 implies b3 is constant )
proof end;

theorem Th30: :: BHSP_3:30
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is constant & b3 is subsequence of b2 implies b2 = b3 )
proof end;

theorem Th31: :: BHSP_3:31
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is bounded & b3 is subsequence of b2 implies b3 is bounded )
proof end;

theorem Th32: :: BHSP_3:32
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & b3 is subsequence of b2 implies b3 is convergent )
proof end;

theorem Th33: :: BHSP_3:33
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is subsequence of b3 & b3 is convergent implies lim b2 = lim b3 )
proof end;

theorem Th34: :: BHSP_3:34
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_Cauchy_sequence & b3 is subsequence of b2 implies b3 is_Cauchy_sequence )
proof end;

definition
let c1 be RealUnitarySpace;
let c2 be sequence of c1;
let c3 be Nat;
func c2 ^\ c3 -> sequence of a1 means :Def5: :: BHSP_3:def 5
for b1 being Nat holds a4 . b1 = a2 . (b1 + a3);
existence
ex b1 being sequence of c1 st
for b2 being Nat holds b1 . b2 = c2 . (b2 + c3)
proof end;
uniqueness
for b1, b2 being sequence of c1 holds
( ( for b3 being Nat holds b1 . b3 = c2 . (b3 + c3) ) & ( for b3 being Nat holds b2 . b3 = c2 . (b3 + c3) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def5 defines ^\ BHSP_3:def 5 :
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat
for b4 being sequence of b1 holds
( b4 = b2 ^\ b3 iff for b5 being Nat holds b4 . b5 = b2 . (b5 + b3) );

theorem Th35: :: BHSP_3:35
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds b2 ^\ 0 = b2
proof end;

theorem Th36: :: BHSP_3:36
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3, b4 being Nat holds (b2 ^\ b3) ^\ b4 = (b2 ^\ b4) ^\ b3
proof end;

theorem Th37: :: BHSP_3:37
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3, b4 being Nat holds (b2 ^\ b3) ^\ b4 = b2 ^\ (b3 + b4)
proof end;

theorem Th38: :: BHSP_3:38
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1
for b4 being Nat holds (b2 + b3) ^\ b4 = (b2 ^\ b4) + (b3 ^\ b4)
proof end;

theorem Th39: :: BHSP_3:39
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds (- b2) ^\ b3 = - (b2 ^\ b3)
proof end;

theorem Th40: :: BHSP_3:40
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1
for b4 being Nat holds (b2 - b3) ^\ b4 = (b2 ^\ b4) - (b3 ^\ b4)
proof end;

theorem Th41: :: BHSP_3:41
for b1 being RealUnitarySpace
for b2 being Real
for b3 being sequence of b1
for b4 being Nat holds (b2 * b3) ^\ b4 = b2 * (b3 ^\ b4)
proof end;

theorem Th42: :: BHSP_3:42
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being increasing Seq_of_Nat
for b4 being Nat holds (b2 * b3) ^\ b4 = b2 * (b3 ^\ b4)
proof end;

theorem Th43: :: BHSP_3:43
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
b2 ^\ b3 is subsequence of b2
proof end;

theorem Th44: :: BHSP_3:44
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
( b2 is convergent implies ( b2 ^\ b3 is convergent & lim (b2 ^\ b3) = lim b2 ) )
proof end;

theorem Th45: :: BHSP_3:45
canceled;

theorem Th46: :: BHSP_3:46
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is convergent & ex b4 being Nat st b2 = b3 ^\ b4 implies b3 is convergent )
proof end;

theorem Th47: :: BHSP_3:47
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1 holds
( b2 is_Cauchy_sequence & ex b4 being Nat st b2 = b3 ^\ b4 implies b3 is_Cauchy_sequence )
proof end;

theorem Th48: :: BHSP_3:48
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
( b2 is_Cauchy_sequence implies b2 ^\ b3 is_Cauchy_sequence )
proof end;

theorem Th49: :: BHSP_3:49
for b1 being RealUnitarySpace
for b2, b3 being sequence of b1
for b4 being Nat holds
( b2 is_compared_to b3 implies b2 ^\ b4 is_compared_to b3 ^\ b4 )
proof end;

theorem Th50: :: BHSP_3:50
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
( b2 is bounded implies b2 ^\ b3 is bounded )
proof end;

theorem Th51: :: BHSP_3:51
for b1 being RealUnitarySpace
for b2 being sequence of b1
for b3 being Nat holds
( b2 is constant implies b2 ^\ b3 is constant )
proof end;

definition
let c1 be RealUnitarySpace;
attr a1 is complete means :Def6: :: BHSP_3:def 6
for b1 being sequence of a1 holds
( b1 is_Cauchy_sequence implies b1 is convergent );
end;

:: deftheorem Def6 defines complete BHSP_3:def 6 :
for b1 being RealUnitarySpace holds
( b1 is complete iff for b2 being sequence of b1 holds
( b2 is_Cauchy_sequence implies b2 is convergent ) );

notation
let c1 be RealUnitarySpace;
synonym c1 is_complete_space for complete c1;
end;

theorem Th52: :: BHSP_3:52
canceled;

theorem Th53: :: BHSP_3:53
for b1 being RealUnitarySpace
for b2 being sequence of b1 holds
( b1 is_complete_space & b2 is_Cauchy_sequence implies b2 is bounded )
proof end;

definition
let c1 be RealUnitarySpace;
attr a1 is Hilbert means :: BHSP_3:def 7
( a1 is RealUnitarySpace & a1 is_complete_space );
end;

:: deftheorem Def7 defines Hilbert BHSP_3:def 7 :
for b1 being RealUnitarySpace holds
( b1 is Hilbert iff ( b1 is RealUnitarySpace & b1 is_complete_space ) );

notation
let c1 be RealUnitarySpace;
synonym c1 is_Hilbert_space for Hilbert c1;
end;