:: TREAL_1 semantic presentation
theorem Th1: :: TREAL_1:1
theorem Th2: :: TREAL_1:2
theorem Th3: :: TREAL_1:3
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 ) )
Lemma6:
for b1, b2 being real number
for b3 being Point of (Closed-Interval-TSpace b1,b2) holds
( b1 <= b2 implies b3 is Real )
theorem Th4: :: TREAL_1:4
theorem Th5: :: TREAL_1:5
theorem Th6: :: TREAL_1:6
theorem Th7: :: TREAL_1:7
:: deftheorem Def1 defines (#) TREAL_1:def 1 :
:: deftheorem Def2 defines (#) TREAL_1:def 2 :
theorem Th8: :: TREAL_1:8
theorem Th9: :: TREAL_1:9
definition
let c
1, c
2 be
real number ;
assume E11:
c
1 <= c
2
;
let c
3, c
4 be
Point of
(Closed-Interval-TSpace c1,c2);
func L[01] c
3,c
4 -> Function of
(Closed-Interval-TSpace 0,1),
(Closed-Interval-TSpace a1,a2) means :
Def3:
:: TREAL_1:def 3
for b
1 being
Point of
(Closed-Interval-TSpace 0,1)for b
2, b
3, b
4 being
real number holds
( b
1 = b
2 & b
3 = a
3 & b
4 = a
4 implies a
5 . b
1 = ((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) )
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 )
end;
:: deftheorem Def3 defines L[01] TREAL_1:def 3 :
for b
1, b
2 being
real number holds
( b
1 <= b
2 implies for b
3, b
4 being
Point of
(Closed-Interval-TSpace b1,b2)for b
5 being
Function of
(Closed-Interval-TSpace 0,1),
(Closed-Interval-TSpace b1,b2) holds
( b
5 = L[01] b
3,b
4 iff for b
6 being
Point of
(Closed-Interval-TSpace 0,1)for b
7, b
8, b
9 being
real number holds
( b
6 = b
7 & b
8 = b
3 & b
9 = b
4 implies b
5 . b
6 = ((1 - b7) * b8) + (b7 * b9) ) ) );
theorem Th10: :: TREAL_1:10
theorem Th11: :: TREAL_1:11
theorem Th12: :: TREAL_1:12
theorem Th13: :: TREAL_1:13
definition
let c
1, c
2 be
real number ;
assume E14:
c
1 < c
2
;
let c
3, c
4 be
Point of
(Closed-Interval-TSpace 0,1);
func P[01] c
1,c
2,c
3,c
4 -> Function of
(Closed-Interval-TSpace a1,a2),
(Closed-Interval-TSpace 0,1) means :
Def4:
:: TREAL_1:def 4
for b
1 being
Point of
(Closed-Interval-TSpace a1,a2)for b
2, b
3, b
4 being
real number holds
( b
1 = b
2 & b
3 = a
3 & b
4 = a
4 implies a
5 . b
1 = (((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) )
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 )
end;
:: deftheorem Def4 defines P[01] TREAL_1:def 4 :
for b
1, b
2 being
real number holds
( b
1 < b
2 implies for b
3, b
4 being
Point of
(Closed-Interval-TSpace 0,1)for b
5 being
Function of
(Closed-Interval-TSpace b1,b2),
(Closed-Interval-TSpace 0,1) holds
( b
5 = P[01] b
1,b
2,b
3,b
4 iff for b
6 being
Point of
(Closed-Interval-TSpace b1,b2)for b
7, b
8, b
9 being
real number holds
( b
6 = b
7 & b
8 = b
3 & b
9 = b
4 implies b
5 . b
6 = (((b2 - b7) * b8) + ((b7 - b1) * b9)) / (b2 - b1) ) ) );
theorem Th14: :: TREAL_1:14
theorem Th15: :: TREAL_1:15
theorem Th16: :: TREAL_1:16
for b
1, b
2 being
real number holds
( b
1 < b
2 implies for b
3, b
4 being
Point of
(Closed-Interval-TSpace 0,1) holds
(
(P[01] b1,b2,b3,b4) . ((#) b1,b2) = b
3 &
(P[01] b1,b2,b3,b4) . (b1,b2 (#) ) = b
4 ) )
theorem Th17: :: TREAL_1:17
theorem Th18: :: TREAL_1:18
for b
1, b
2 being
real number holds
( b
1 < b
2 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 (#) )) ) )
theorem Th19: :: TREAL_1:19
for b
1, b
2 being
real number holds
( b
1 < b
2 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)) ) )
theorem Th20: :: TREAL_1:20
for b
1, b
2 being
real number holds
( b
1 < b
2 implies (
L[01] ((#) b1,b2),
(b1,b2 (#) ) is_homeomorphism &
(L[01] ((#) b1,b2),(b1,b2 (#) )) " = P[01] b
1,b
2,
((#) 0,1),
(0,1 (#) ) &
P[01] b
1,b
2,
((#) 0,1),
(0,1 (#) ) is_homeomorphism &
(P[01] b1,b2,((#) 0,1),(0,1 (#) )) " = L[01] ((#) b1,b2),
(b1,b2 (#) ) ) )
theorem Th21: :: TREAL_1:21
for b
1, b
2 being
real number holds
( b
1 < b
2 implies (
L[01] (b1,b2 (#) ),
((#) b1,b2) is_homeomorphism &
(L[01] (b1,b2 (#) ),((#) b1,b2)) " = P[01] b
1,b
2,
(0,1 (#) ),
((#) 0,1) &
P[01] b
1,b
2,
(0,1 (#) ),
((#) 0,1) is_homeomorphism &
(P[01] b1,b2,(0,1 (#) ),((#) 0,1)) " = L[01] (b1,b2 (#) ),
((#) b1,b2) ) )
theorem Th22: :: TREAL_1:22
theorem Th23: :: TREAL_1:23
theorem Th24: :: TREAL_1:24
theorem Th25: :: TREAL_1:25
theorem Th26: :: TREAL_1:26
theorem Th27: :: TREAL_1:27