:: TREAL_1 semantic presentation
theorem Th1: :: TREAL_1:1
theorem Th2: :: TREAL_1:2
theorem Th3: :: TREAL_1:3
Lemma28:
for a, b being real number
for x being set st x in [.a,b.] holds
x is Real
;
Lemma30:
for a, b, v being real number
for x being set st x in [.a,b.] & a <= b & v = x holds
( a <= v & v <= b )
Lemma35:
for a, b being real number
for x being Point of (Closed-Interval-TSpace a,b) st a <= b holds
x 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 a be
real number ,
b be
real number ;
assume E32:
a <= b
;
let t1 be
Point of
(Closed-Interval-TSpace a,b),
t2 be
Point of
(Closed-Interval-TSpace a,b);
func L[01] c3,
c4 -> Function of
(Closed-Interval-TSpace 0,1),
(Closed-Interval-TSpace a1,a2) means :
Def3:
:: TREAL_1:def 3
for
s being
Point of
(Closed-Interval-TSpace 0,1) for
r,
r1,
r2 being
real number st
s = r &
r1 = t1 &
r2 = t2 holds
it . s = ((1 - r) * r1) + (r * r2);
existence
ex b1 being Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace a,b) st
for s being Point of (Closed-Interval-TSpace 0,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b1 . s = ((1 - r) * r1) + (r * r2)
uniqueness
for b1, b2 being Function of (Closed-Interval-TSpace 0,1),(Closed-Interval-TSpace a,b) st ( for s being Point of (Closed-Interval-TSpace 0,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b1 . s = ((1 - r) * r1) + (r * r2) ) & ( for s being Point of (Closed-Interval-TSpace 0,1)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b2 . s = ((1 - r) * r1) + (r * r2) ) holds
b1 = b2
end;
:: deftheorem Def3 defines L[01] TREAL_1:def 3 :
for
a,
b being
real number st
a <= b holds
for
t1,
t2 being
Point of
(Closed-Interval-TSpace a,b) for
b5 being
Function of
(Closed-Interval-TSpace 0,1),
(Closed-Interval-TSpace a,b) holds
(
b5 = L[01] t1,
t2 iff for
s being
Point of
(Closed-Interval-TSpace 0,1) for
r,
r1,
r2 being
real number st
s = r &
r1 = t1 &
r2 = t2 holds
b5 . s = ((1 - r) * r1) + (r * r2) );
theorem Th10: :: TREAL_1:10
theorem Th11: :: TREAL_1:11
theorem Th12: :: TREAL_1:12
theorem Th13: :: TREAL_1:13
definition
let a be
real number ,
b be
real number ;
assume E32:
a < b
;
let t1 be
Point of
(Closed-Interval-TSpace 0,1),
t2 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
s being
Point of
(Closed-Interval-TSpace a,b) for
r,
r1,
r2 being
real number st
s = r &
r1 = t1 &
r2 = t2 holds
it . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a);
existence
ex b1 being Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0,1) st
for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b1 . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a)
uniqueness
for b1, b2 being Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace 0,1) st ( for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b1 . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) ) & ( for s being Point of (Closed-Interval-TSpace a,b)
for r, r1, r2 being real number st s = r & r1 = t1 & r2 = t2 holds
b2 . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) ) holds
b1 = b2
end;
:: deftheorem Def4 defines P[01] TREAL_1:def 4 :
for
a,
b being
real number st
a < b holds
for
t1,
t2 being
Point of
(Closed-Interval-TSpace 0,1) for
b5 being
Function of
(Closed-Interval-TSpace a,b),
(Closed-Interval-TSpace 0,1) holds
(
b5 = P[01] a,
b,
t1,
t2 iff for
s being
Point of
(Closed-Interval-TSpace a,b) for
r,
r1,
r2 being
real number st
s = r &
r1 = t1 &
r2 = t2 holds
b5 . s = (((b - r) * r1) + ((r - a) * r2)) / (b - a) );
theorem Th14: :: TREAL_1:14
theorem Th15: :: TREAL_1:15
theorem Th16: :: TREAL_1:16
for
a,
b being
real number st
a < b holds
for
t1,
t2 being
Point of
(Closed-Interval-TSpace 0,1) holds
(
(P[01] a,b,t1,t2) . ((#) a,b) = t1 &
(P[01] a,b,t1,t2) . (a,b (#) ) = t2 )
theorem Th17: :: TREAL_1:17
theorem Th18: :: TREAL_1:18
for
a,
b being
real number st
a < b holds
(
id (Closed-Interval-TSpace a,b) = (L[01] ((#) a,b),(a,b (#) )) * (P[01] a,b,((#) 0,1),(0,1 (#) )) &
id (Closed-Interval-TSpace 0,1) = (P[01] a,b,((#) 0,1),(0,1 (#) )) * (L[01] ((#) a,b),(a,b (#) )) )
theorem Th19: :: TREAL_1:19
for
a,
b being
real number st
a < b holds
(
id (Closed-Interval-TSpace a,b) = (L[01] (a,b (#) ),((#) a,b)) * (P[01] a,b,(0,1 (#) ),((#) 0,1)) &
id (Closed-Interval-TSpace 0,1) = (P[01] a,b,(0,1 (#) ),((#) 0,1)) * (L[01] (a,b (#) ),((#) a,b)) )
theorem Th20: :: TREAL_1:20
for
a,
b being
real number st
a < b holds
(
L[01] ((#) a,b),
(a,b (#) ) is_homeomorphism &
(L[01] ((#) a,b),(a,b (#) )) " = P[01] a,
b,
((#) 0,1),
(0,1 (#) ) &
P[01] a,
b,
((#) 0,1),
(0,1 (#) ) is_homeomorphism &
(P[01] a,b,((#) 0,1),(0,1 (#) )) " = L[01] ((#) a,b),
(a,b (#) ) )
theorem Th21: :: TREAL_1:21
for
a,
b being
real number st
a < b holds
(
L[01] (a,b (#) ),
((#) a,b) is_homeomorphism &
(L[01] (a,b (#) ),((#) a,b)) " = P[01] a,
b,
(0,1 (#) ),
((#) 0,1) &
P[01] a,
b,
(0,1 (#) ),
((#) 0,1) is_homeomorphism &
(P[01] a,b,(0,1 (#) ),((#) 0,1)) " = L[01] (a,b (#) ),
((#) a,b) )
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