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;