reserve a,b,c,d for Real;

theorem Th17:
  a < b implies
  L[01]((#)(a,b),(a,b)(#)) is being_homeomorphism &
  L[01]((#)(a,b),(a,b)(#))" = P[01](a,b,(#)(0,1),(0,1)(#)) &
  P[01](a,b,(#)(0,1),(0,1)(#)) is being_homeomorphism &
  P[01](a,b,(#)(0,1),(0,1)(#))" = L[01]((#)(a,b),(a,b)(#))
proof
  set L = L[01]((#)(a,b),(a,b)(#)), P = P[01](a,b,(#)(0,1),(0,1)(#));
  assume
A1: a < b;
  then
A2: id (the carrier of Closed-Interval-TSpace(0,1)) = P * L by Th15;
  then
A3: L is one-to-one by FUNCT_2:23;
A4: L is continuous & P is continuous Function of Closed-Interval-TSpace(a,
  b), Closed-Interval-TSpace(0,1) by A1,Th8,Th12;
A5: id (the carrier of Closed-Interval-TSpace(a,b)) = id
  Closed-Interval-TSpace(a,b)
    .= L * P by A1,Th15;
  then
A6: L is onto by FUNCT_2:23;
  then
A7: rng L = [#](Closed-Interval-TSpace(a,b));
A8: L" = L qua Function" by A3,A6,TOPS_2:def 4;
  dom L = [#]Closed-Interval-TSpace(0,1) & P = L qua Function" by A2,A3,A7,
FUNCT_2:30,def 1;
  hence L[01]((#)(a,b),(a,b)(#)) is being_homeomorphism by A3,A7,A8,A4,
TOPS_2:def 5;
  thus L[01]((#)(a,b),(a,b)(#))" = P[01](a,b,(#)(0,1),(0,1)(#)) by A2,A3,A7,A8,
FUNCT_2:30;
A9:  P is onto by A2,FUNCT_2:23;
  then
A10: rng P = [#](Closed-Interval-TSpace(0,1));
A11: L is continuous Function of Closed-Interval-TSpace(0,1),
  Closed-Interval-TSpace(a,b) & P is continuous by A1,Th8,Th12;
A12: P is one-to-one by A5,FUNCT_2:23;
A13: P" = P qua Function" by A12,A9,TOPS_2:def 4;
  dom P = [#]Closed-Interval-TSpace(a,b) & L = P qua Function" by A10,A5,A12,
FUNCT_2:30,def 1;
  hence P[01](a,b,(#)(0,1),(0,1)(#)) is being_homeomorphism by A10,A12,A13,A11,
TOPS_2:def 5;
  thus thesis by A10,A5,A12,A13,FUNCT_2:30;
end;
