:: TREAL_1 semantic presentation

theorem Th1: :: TREAL_1:1
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 implies [.b2,b3.] c= [.b1,b4.] )
proof end;

theorem Th2: :: TREAL_1:2
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 & b2 <= b3 implies [.b1,b3.] \/ [.b2,b4.] = [.b1,b4.] )
proof end;

theorem Th3: :: TREAL_1:3
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 & b2 <= b3 implies [.b1,b3.] /\ [.b2,b4.] = [.b2,b3.] )
proof end;

Lemma4: for b1, b2 being real number
for b3 being set holds
( b3 in [.b1,b2.] implies b3 is Real )
;

Lemma5: for b1, b2, b3 being real number
for b4 being set holds
( b4 in [.b1,b2.] & b1 <= b2 & b3 = b4 implies ( b1 <= b3 & b3 <= b2 ) )
proof end;

Lemma6: for b1, b2 being real number
for b3 being Point of (Closed-Interval-TSpace b1,b2) holds
( b1 <= b2 implies b3 is Real )
proof end;

theorem Th4: :: TREAL_1:4
for b1, b2 being real number
for b3 being Subset of R^1 holds
( b3 = [.b1,b2.] implies b3 is closed )
proof end;

theorem Th5: :: TREAL_1:5
for b1, b2 being real number holds
( b1 <= b2 implies Closed-Interval-TSpace b1,b2 is closed SubSpace of R^1 )
proof end;

theorem Th6: :: TREAL_1:6
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 & b2 <= b3 implies Closed-Interval-TSpace b2,b3 is closed SubSpace of Closed-Interval-TSpace b1,b4 )
proof end;

theorem Th7: :: TREAL_1:7
for b1, b2, b3, b4 being real number holds
( b1 <= b2 & b3 <= b4 & b2 <= b3 implies ( Closed-Interval-TSpace b1,b4 = (Closed-Interval-TSpace b1,b3) union (Closed-Interval-TSpace b2,b4) & Closed-Interval-TSpace b2,b3 = (Closed-Interval-TSpace b1,b3) meet (Closed-Interval-TSpace b2,b4) ) )
proof end;

definition
let c1, c2 be real number ;
assume E9: c1 <= c2 ;
func (#) c1,c2 -> Point of (Closed-Interval-TSpace a1,a2) equals :Def1: :: TREAL_1:def 1
a1;
coherence
c1 is Point of (Closed-Interval-TSpace c1,c2)
proof end;
correctness
;
func c1,c2 (#) -> Point of (Closed-Interval-TSpace a1,a2) equals :Def2: :: TREAL_1:def 2
a2;
coherence
c2 is Point of (Closed-Interval-TSpace c1,c2)
proof end;
correctness
;
end;

:: deftheorem Def1 defines (#) TREAL_1:def 1 :
for b1, b2 being real number holds
( b1 <= b2 implies (#) b1,b2 = b1 );

:: deftheorem Def2 defines (#) TREAL_1:def 2 :
for b1, b2 being real number holds
( b1 <= b2 implies b1,b2 (#) = b2 );

theorem Th8: :: TREAL_1:8
( 0[01] = (#) 0,1 & 1[01] = 0,1 (#) ) by Def1, Def2, BORSUK_1:def 17, BORSUK_1:def 18;

theorem Th9: :: TREAL_1:9
for b1, b2, b3 being real number holds
( b1 <= b2 & b2 <= b3 implies ( (#) b1,b2 = (#) b1,b3 & b2,b3 (#) = b1,b3 (#) ) )
proof end;

definition
let c1, c2 be real number ;
assume E11: c1 <= c2 ;
let c3, c4 be Point of (Closed-Interval-TSpace c1,c2);
func L[01] c3,c4 -> Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace a1,a2) means :Def3: :: TREAL_1:def 3
for b1 being Point of (Closed-Interval-TSpace 0,1)
for b2, b3, b4 being real number holds
( b1 = b2 & b3 = a3 & b4 = a4 implies a5 . b1 = ((1 - b2) * b3) + (b2 * b4) );
existence
ex b1 being Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace c1,c2) st
for b2 being Point of (Closed-Interval-TSpace 0,1)
for b3, b4, b5 being real number holds
( b2 = b3 & b4 = c3 & b5 = c4 implies b1 . b2 = ((1 - b3) * b4) + (b3 * b5) )
proof end;
uniqueness
for b1, b2 being Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace c1,c2) holds
( ( for b3 being Point of (Closed-Interval-TSpace 0,1)
for b4, b5, b6 being real number holds
( b3 = b4 & b5 = c3 & b6 = c4 implies b1 . b3 = ((1 - b4) * b5) + (b4 * b6) ) ) & ( for b3 being Point of (Closed-Interval-TSpace 0,1)
for b4, b5, b6 being real number holds
( b3 = b4 & b5 = c3 & b6 = c4 implies b2 . b3 = ((1 - b4) * b5) + (b4 * b6) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def3 defines L[01] TREAL_1:def 3 :
for b1, b2 being real number holds
( b1 <= b2 implies for b3, b4 being Point of (Closed-Interval-TSpace b1,b2)
for b5 being Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace b1,b2) holds
( b5 = L[01] b3,b4 iff for b6 being Point of (Closed-Interval-TSpace 0,1)
for b7, b8, b9 being real number holds
( b6 = b7 & b8 = b3 & b9 = b4 implies b5 . b6 = ((1 - b7) * b8) + (b7 * b9) ) ) );

theorem Th10: :: TREAL_1:10
for b1, b2 being real number holds
( b1 <= b2 implies for b3, b4 being Point of (Closed-Interval-TSpace b1,b2)
for b5 being Point of (Closed-Interval-TSpace 0,1)
for b6, b7, b8 being real number holds
( b5 = b6 & b7 = b3 & b8 = b4 implies (L[01] b3,b4) . b5 = ((b8 - b7) * b6) + b7 ) )
proof end;

theorem Th11: :: TREAL_1:11
for b1, b2 being real number holds
( b1 <= b2 implies for b3, b4 being Point of (Closed-Interval-TSpace b1,b2) holds
L[01] b3,b4 is continuous Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace b1,b2) )
proof end;

theorem Th12: :: TREAL_1:12
for b1, b2 being real number holds
( b1 <= b2 implies for b3, b4 being Point of (Closed-Interval-TSpace b1,b2) holds
( (L[01] b3,b4) . ((#) 0,1) = b3 & (L[01] b3,b4) . (0,1 (#) ) = b4 ) )
proof end;

theorem Th13: :: TREAL_1:13
L[01] ((#) 0,1),(0,1 (#) ) = id (Closed-Interval-TSpace 0,1)
proof end;

definition
let c1, c2 be real number ;
assume E14: c1 < c2 ;
let c3, c4 be Point of (Closed-Interval-TSpace 0,1);
func P[01] c1,c2,c3,c4 -> Function of (Closed-Interval-TSpace a1,a2),(Closed-Interval-TSpace 0,1) means :Def4: :: TREAL_1:def 4
for b1 being Point of (Closed-Interval-TSpace a1,a2)
for b2, b3, b4 being real number holds
( b1 = b2 & b3 = a3 & b4 = a4 implies a5 . b1 = (((a2 - b2) * b3) + ((b2 - a1) * b4)) / (a2 - a1) );
existence
ex b1 being Function of (Closed-Interval-TSpace c1,c2),(Closed-Interval-TSpace 0,1) st
for b2 being Point of (Closed-Interval-TSpace c1,c2)
for b3, b4, b5 being real number holds
( b2 = b3 & b4 = c3 & b5 = c4 implies b1 . b2 = (((c2 - b3) * b4) + ((b3 - c1) * b5)) / (c2 - c1) )
proof end;
uniqueness
for b1, b2 being Function of (Closed-Interval-TSpace c1,c2),(Closed-Interval-TSpace 0,1) holds
( ( for b3 being Point of (Closed-Interval-TSpace c1,c2)
for b4, b5, b6 being real number holds
( b3 = b4 & b5 = c3 & b6 = c4 implies b1 . b3 = (((c2 - b4) * b5) + ((b4 - c1) * b6)) / (c2 - c1) ) ) & ( for b3 being Point of (Closed-Interval-TSpace c1,c2)
for b4, b5, b6 being real number holds
( b3 = b4 & b5 = c3 & b6 = c4 implies b2 . b3 = (((c2 - b4) * b5) + ((b4 - c1) * b6)) / (c2 - c1) ) ) implies b1 = b2 )
proof end;
end;

:: deftheorem Def4 defines P[01] TREAL_1:def 4 :
for b1, b2 being real number holds
( b1 < b2 implies for b3, b4 being Point of (Closed-Interval-TSpace 0,1)
for b5 being Function of (Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace 0,1) holds
( b5 = P[01] b1,b2,b3,b4 iff for b6 being Point of (Closed-Interval-TSpace b1,b2)
for b7, b8, b9 being real number holds
( b6 = b7 & b8 = b3 & b9 = b4 implies b5 . b6 = (((b2 - b7) * b8) + ((b7 - b1) * b9)) / (b2 - b1) ) ) );

theorem Th14: :: TREAL_1:14
for b1, b2 being real number holds
( b1 < b2 implies for b3, b4 being Point of (Closed-Interval-TSpace 0,1)
for b5 being Point of (Closed-Interval-TSpace b1,b2)
for b6, b7, b8 being Real holds
( b5 = b6 & b7 = b3 & b8 = b4 implies (P[01] b1,b2,b3,b4) . b5 = (((b8 - b7) / (b2 - b1)) * b6) + (((b2 * b7) - (b1 * b8)) / (b2 - b1)) ) )
proof end;

theorem Th15: :: TREAL_1:15
for b1, b2 being real number holds
( b1 < b2 implies for b3, b4 being Point of (Closed-Interval-TSpace 0,1) holds
P[01] b1,b2,b3,b4 is continuous Function of (Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace 0,1) )
proof end;

theorem Th16: :: TREAL_1:16
for b1, b2 being real number holds
( b1 < b2 implies for b3, b4 being Point of (Closed-Interval-TSpace 0,1) holds
( (P[01] b1,b2,b3,b4) . ((#) b1,b2) = b3 & (P[01] b1,b2,b3,b4) . (b1,b2 (#) ) = b4 ) )
proof end;

theorem Th17: :: TREAL_1:17
P[01] 0,1,((#) 0,1),(0,1 (#) ) = id (Closed-Interval-TSpace 0,1)
proof end;

theorem Th18: :: TREAL_1:18
for b1, b2 being real number holds
( b1 < b2 implies ( id (Closed-Interval-TSpace b1,b2) = (L[01] ((#) b1,b2),(b1,b2 (#) )) * (P[01] b1,b2,((#) 0,1),(0,1 (#) )) & id (Closed-Interval-TSpace 0,1) = (P[01] b1,b2,((#) 0,1),(0,1 (#) )) * (L[01] ((#) b1,b2),(b1,b2 (#) )) ) )
proof end;

theorem Th19: :: TREAL_1:19
for b1, b2 being real number holds
( b1 < b2 implies ( id (Closed-Interval-TSpace b1,b2) = (L[01] (b1,b2 (#) ),((#) b1,b2)) * (P[01] b1,b2,(0,1 (#) ),((#) 0,1)) & id (Closed-Interval-TSpace 0,1) = (P[01] b1,b2,(0,1 (#) ),((#) 0,1)) * (L[01] (b1,b2 (#) ),((#) b1,b2)) ) )
proof end;

theorem Th20: :: TREAL_1:20
for b1, b2 being real number holds
( b1 < b2 implies ( L[01] ((#) b1,b2),(b1,b2 (#) ) is_homeomorphism & (L[01] ((#) b1,b2),(b1,b2 (#) )) " = P[01] b1,b2,((#) 0,1),(0,1 (#) ) & P[01] b1,b2,((#) 0,1),(0,1 (#) ) is_homeomorphism & (P[01] b1,b2,((#) 0,1),(0,1 (#) )) " = L[01] ((#) b1,b2),(b1,b2 (#) ) ) )
proof end;

theorem Th21: :: TREAL_1:21
for b1, b2 being real number holds
( b1 < b2 implies ( L[01] (b1,b2 (#) ),((#) b1,b2) is_homeomorphism & (L[01] (b1,b2 (#) ),((#) b1,b2)) " = P[01] b1,b2,(0,1 (#) ),((#) 0,1) & P[01] b1,b2,(0,1 (#) ),((#) 0,1) is_homeomorphism & (P[01] b1,b2,(0,1 (#) ),((#) 0,1)) " = L[01] (b1,b2 (#) ),((#) b1,b2) ) )
proof end;

theorem Th22: :: TREAL_1:22
I[01] is connected
proof end;

theorem Th23: :: TREAL_1:23
for b1, b2 being real number holds
( b1 <= b2 implies Closed-Interval-TSpace b1,b2 is connected )
proof end;

theorem Th24: :: TREAL_1:24
for b1 being continuous Function of I[01] ,I[01] holds
ex b2 being Point of I[01] st b1 . b2 = b2
proof end;

theorem Th25: :: TREAL_1:25
for b1, b2 being real number holds
( b1 <= b2 implies for b3 being continuous Function of (Closed-Interval-TSpace b1,b2),(Closed-Interval-TSpace b1,b2) holds
ex b4 being Point of (Closed-Interval-TSpace b1,b2) st b3 . b4 = b4 )
proof end;

theorem Th26: :: TREAL_1:26
for b1, b2 being non empty SubSpace of R^1
for b3 being continuous Function of b1,b2 holds
not ( ex b4, b5 being Real st
( b4 <= b5 & [.b4,b5.] c= the carrier of b1 & [.b4,b5.] c= the carrier of b2 & b3 .: [.b4,b5.] c= [.b4,b5.] ) & ( for b4 being Point of b1 holds
not b3 . b4 = b4 ) )
proof end;

theorem Th27: :: TREAL_1:27
for b1, b2 being non empty SubSpace of R^1
for b3 being continuous Function of b1,b2 holds
not ( ex b4, b5 being Real st
( b4 <= b5 & [.b4,b5.] c= the carrier of b1 & b3 .: [.b4,b5.] c= [.b4,b5.] ) & ( for b4 being Point of b1 holds
not b3 . b4 = b4 ) )
proof end;