Article TREAL_1, MML version 4.99.1005
:: TREAL_1:th 4
theorem
for b1, b2 being real set
for b3 being Element of bool the carrier of R^1
st b3 = [.b1,b2.]
holds b3 is closed(R^1);
:: TREAL_1:th 5
theorem
for b1, b2 being real set
st b1 <= b2
holds Closed-Interval-TSpace(b1,b2) is closed SubSpace of R^1;
:: TREAL_1:th 6
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4 & b2 <= b3
holds Closed-Interval-TSpace(b2,b3) is closed SubSpace of Closed-Interval-TSpace(b1,b4);
:: TREAL_1:th 7
theorem
for b1, b2, b3, b4 being real set
st b1 <= b2 & b3 <= b4 & b2 <= b3
holds 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);
:: TREAL_1:funcnot 1 => TREAL_1:func 1
definition
let a1, a2 be real set;
assume a1 <= a2;
func (#)(A1,A2) -> Element of the carrier of Closed-Interval-TSpace(a1,a2) equals
a1;
end;
:: TREAL_1:def 1
theorem
for b1, b2 being real set
st b1 <= b2
holds (#)(b1,b2) = b1;
:: TREAL_1:funcnot 2 => TREAL_1:func 2
definition
let a1, a2 be real set;
assume a1 <= a2;
func (A1,A2)(#) -> Element of the carrier of Closed-Interval-TSpace(a1,a2) equals
a2;
end;
:: TREAL_1:def 2
theorem
for b1, b2 being real set
st b1 <= b2
holds (b1,b2)(#) = b2;
:: TREAL_1:th 8
theorem
0[01] = (#)(0,1) & 1[01] = (0,1)(#);
:: TREAL_1:th 9
theorem
for b1, b2, b3 being real set
st b1 <= b2 & b2 <= b3
holds (#)(b1,b2) = (#)(b1,b3) & (b2,b3)(#) = (b1,b3)(#);
:: TREAL_1:funcnot 3 => TREAL_1:func 3
definition
let a1, a2 be real set;
let a3, a4 be Element of the carrier of Closed-Interval-TSpace(a1,a2);
assume a1 <= a2;
func L[01](A3,A4) -> Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(0,1),the carrier of Closed-Interval-TSpace(a1,a2) means
for b1 being Element of the carrier of Closed-Interval-TSpace(0,1)
for b2, b3, b4 being real set
st b1 = b2 & b3 = a3 & b4 = a4
holds it . b1 = ((1 - b2) * b3) + (b2 * b4);
end;
:: TREAL_1:def 3
theorem
for b1, b2 being real set
st b1 <= b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(0,1),the carrier of Closed-Interval-TSpace(b1,b2) holds
b5 = L[01](b3,b4)
iff
for b6 being Element of the carrier of Closed-Interval-TSpace(0,1)
for b7, b8, b9 being real set
st b6 = b7 & b8 = b3 & b9 = b4
holds b5 . b6 = ((1 - b7) * b8) + (b7 * b9);
:: TREAL_1:th 10
theorem
for b1, b2 being real set
st b1 <= b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b5 being Element of the carrier of Closed-Interval-TSpace(0,1)
for b6, b7, b8 being real set
st b5 = b6 & b7 = b3 & b8 = b4
holds (L[01](b3,b4)) . b5 = ((b8 - b7) * b6) + b7;
:: TREAL_1:th 11
theorem
for b1, b2 being real set
st b1 <= b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(b1,b2) holds
L[01](b3,b4) is Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(0,1),the carrier of Closed-Interval-TSpace(b1,b2);
:: TREAL_1:th 12
theorem
for b1, b2 being real set
st b1 <= b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(b1,b2) holds
(L[01](b3,b4)) . (#)(0,1) = b3 &
(L[01](b3,b4)) . ((0,1)(#)) = b4;
:: TREAL_1:th 13
theorem
L[01]((#)(0,1),(0,1)(#)) = id Closed-Interval-TSpace(0,1);
:: TREAL_1:funcnot 4 => TREAL_1:func 4
definition
let a1, a2 be real set;
let a3, a4 be Element of the carrier of Closed-Interval-TSpace(0,1);
assume a1 < a2;
func P[01](A1,A2,A3,A4) -> Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(a1,a2),the carrier of Closed-Interval-TSpace(0,1) means
for b1 being Element of the carrier of Closed-Interval-TSpace(a1,a2)
for b2, b3, b4 being real set
st b1 = b2 & b3 = a3 & b4 = a4
holds it . b1 = (((a2 - b2) * b3) + ((b2 - a1) * b4)) / (a2 - a1);
end;
:: TREAL_1:def 4
theorem
for b1, b2 being real set
st b1 < b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(0,1)
for b5 being Function-like quasi_total Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(0,1) holds
b5 = P[01](b1,b2,b3,b4)
iff
for b6 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b7, b8, b9 being real set
st b6 = b7 & b8 = b3 & b9 = b4
holds b5 . b6 = (((b2 - b7) * b8) + ((b7 - b1) * b9)) / (b2 - b1);
:: TREAL_1:th 14
theorem
for b1, b2 being real set
st b1 < b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(0,1)
for b5 being Element of the carrier of Closed-Interval-TSpace(b1,b2)
for b6, b7, b8 being Element of REAL
st b5 = b6 & b7 = b3 & b8 = b4
holds (P[01](b1,b2,b3,b4)) . b5 = (((b8 - b7) / (b2 - b1)) * b6) + (((b2 * b7) - (b1 * b8)) / (b2 - b1));
:: TREAL_1:th 15
theorem
for b1, b2 being real set
st b1 < b2
for b3, b4 being Element of the carrier of Closed-Interval-TSpace(0,1) holds
P[01](b1,b2,b3,b4) is Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(0,1);
:: TREAL_1:th 16
theorem
for b1, b2 being real set
st b1 < b2
for b3, b4 being Element of the carrier 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;
:: TREAL_1:th 17
theorem
P[01](0,1,(#)(0,1),(0,1)(#)) = id Closed-Interval-TSpace(0,1);
:: TREAL_1:th 18
theorem
for b1, b2 being real set
st b1 < b2
holds 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)(#));
:: TREAL_1:th 19
theorem
for b1, b2 being real set
st b1 < b2
holds 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));
:: TREAL_1:th 20
theorem
for b1, b2 being real set
st b1 < b2
holds L[01]((#)(b1,b2),(b1,b2)(#)) is being_homeomorphism(Closed-Interval-TSpace(0,1), Closed-Interval-TSpace(b1,b2)) &
(L[01]((#)(b1,b2),(b1,b2)(#))) /" = P[01](b1,b2,(#)(0,1),(0,1)(#)) &
P[01](b1,b2,(#)(0,1),(0,1)(#)) is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(0,1)) &
(P[01](b1,b2,(#)(0,1),(0,1)(#))) /" = L[01]((#)(b1,b2),(b1,b2)(#));
:: TREAL_1:th 21
theorem
for b1, b2 being real set
st b1 < b2
holds L[01]((b1,b2)(#),(#)(b1,b2)) is being_homeomorphism(Closed-Interval-TSpace(0,1), Closed-Interval-TSpace(b1,b2)) &
(L[01]((b1,b2)(#),(#)(b1,b2))) /" = P[01](b1,b2,(0,1)(#),(#)(0,1)) &
P[01](b1,b2,(0,1)(#),(#)(0,1)) is being_homeomorphism(Closed-Interval-TSpace(b1,b2), Closed-Interval-TSpace(0,1)) &
(P[01](b1,b2,(0,1)(#),(#)(0,1))) /" = L[01]((b1,b2)(#),(#)(b1,b2));
:: TREAL_1:th 22
theorem
I[01] is connected;
:: TREAL_1:th 23
theorem
for b1, b2 being real set
st b1 <= b2
holds Closed-Interval-TSpace(b1,b2) is connected;
:: TREAL_1:th 24
theorem
for b1 being Function-like quasi_total continuous Relation of the carrier of I[01],the carrier of I[01] holds
ex b2 being Element of the carrier of I[01] st
b1 . b2 = b2;
:: TREAL_1:th 25
theorem
for b1, b2 being real set
st b1 <= b2
for b3 being Function-like quasi_total continuous Relation of the carrier of Closed-Interval-TSpace(b1,b2),the carrier of Closed-Interval-TSpace(b1,b2) holds
ex b4 being Element of the carrier of Closed-Interval-TSpace(b1,b2) st
b3 . b4 = b4;
:: TREAL_1:th 26
theorem
for b1, b2 being non empty SubSpace of R^1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
st ex b4, b5 being Element of REAL st
b4 <= b5 &
[.b4,b5.] c= the carrier of b1 &
[.b4,b5.] c= the carrier of b2 &
b3 .: [.b4,b5.] c= [.b4,b5.]
holds ex b4 being Element of the carrier of b1 st
b3 . b4 = b4;
:: TREAL_1:th 27
theorem
for b1, b2 being non empty SubSpace of R^1
for b3 being Function-like quasi_total continuous Relation of the carrier of b1,the carrier of b2
st ex b4, b5 being Element of REAL st
b4 <= b5 &
[.b4,b5.] c= the carrier of b1 &
b3 .: [.b4,b5.] c= [.b4,b5.]
holds ex b4 being Element of the carrier of b1 st
b3 . b4 = b4;