
theorem Th16:
  for X,Y be non empty set,
      S be SigmaField of X, T be Function of X,Y,
      f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL,
      A be Element of S, B be Element of CopyField(T,S)
  st T is bijective & g = f*T" holds
   for r be Real holds T.:(less_dom(f,r)) = less_dom(g,r)
proof
    let X,Y be non empty set, S be SigmaField of X, T be Function of X,Y,
    f be PartFunc of X,ExtREAL, g be PartFunc of Y,ExtREAL,
    A be Element of S, B be Element of CopyField(T,S);
    assume that
A1: T is bijective and
A2: g = f*T";

A3: dom T = X & rng T = Y by A1,FUNCT_2:def 1,def 3;
A4: rng T = dom(T") & dom T = rng(T") by A1,FUNCT_1:33;
A5: dom f c= X & rng f c= ExtREAL;

    g*T = f*(T"*T) by A2,RELAT_1:36; then
    g*T = f* (id dom T) by A1,FUNCT_1:39; then
A6: g*T = f by A3,A5,RELAT_1:51;

    let r be Real;

    for x be object holds x in T.:(less_dom(f,r)) iff x in less_dom(g,r)
    proof
     let x be object;
     hereby assume x in T.:(less_dom(f,r)); then
      consider t be object such that
A7:   t in dom T & t in less_dom(f,r) & x= T.t by FUNCT_1:def 6;
A8:   t in dom f & f.t < r by A7,MESFUNC1:def 11;
A9:   g.(T.t) = f.t by A6,A7,FUNCT_1:13;

      T.t in dom g by A8,A6,FUNCT_1:11;
      hence x in less_dom(g,r) by A7,A8,A9,MESFUNC1:def 11;
     end;
     assume
A10:  x in less_dom(g,r); then
      x in dom g & g.x < r by MESFUNC1:def 11; then
     T".x in dom f & f.(T".x) < r by A3,A4,A2,FUNCT_1:11,13; then
A11: T".x in less_dom(f,r) by MESFUNC1:def 11;

     x = T.(T".x) by A1,A10,FUNCT_1:35,A3;
     hence x in T.:(less_dom(f,r)) by A11,A3,FUNCT_1:def 6;
    end;
    hence T.:(less_dom(f,r)) = less_dom(g,r) by TARSKI:2;
end;
