
theorem Th27:
for X,Y be non empty set, T be Function of X,Y, f be PartFunc of X,ExtREAL,
 g be PartFunc of Y,ExtREAL st T is bijective & g = f*T" holds
  max+g = (max+ f)*T" & max-g = (max- f)*T"
proof
    let X,Y be non empty set, T be Function of X,Y, f be PartFunc of X,ExtREAL,
    g be PartFunc of Y,ExtREAL;
    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 (max+ f) = dom f & dom (max-f) = dom f by MESFUNC2:def 2,def 3;

    reconsider H = T" as Function of Y,X by A3,A4,FUNCT_2:1;
    reconsider g1 = (max+f)*H as PartFunc of Y,ExtREAL;

A6: for x be object holds x in dom g1 iff x in dom g
    proof :::
     let x be object;
     hereby assume x in dom g1; then
      x in dom H & H.x in dom (max+f) by FUNCT_1:11;
      hence x in dom g by A2,A5,FUNCT_1:11;
     end;
     assume x in dom g; then
     x in dom H & H.x in dom f by A2,FUNCT_1:11;
     hence x in dom g1 by FUNCT_1:11,A5;
    end;

    for y being Element of Y st y in dom g1 holds g1.y = max((g.y),0.)
    proof
     let y be Element of Y;
     assume y in dom g1; then
A7:  y in dom H & H.y in dom (max+f) by FUNCT_1:11;
     reconsider x= H.y as Element of X;
     g1.y = (max+f). x by FUNCT_1:13,A7; then
     g1.y = max(f.x,0.) by A7,MESFUNC2:def 2;
     hence g1.y = max(g.y,0.) by A2,A7,FUNCT_1:13;
    end;
    hence max+g = (max+f)*T" by A6,TARSKI:2,MESFUNC2:def 2;

    reconsider g1 = (max-f)*H as PartFunc of Y,ExtREAL;

A8: for x be object holds x in dom g1 iff x in dom g
    proof :::
     let x be object;
     hereby assume x in dom g1; then
      x in dom H & H.x in dom (max-f) by FUNCT_1:11;
      hence x in dom g by A2,A5,FUNCT_1:11;
     end;
     assume x in dom g; then
     x in dom H & H.x in dom f by A2,FUNCT_1:11;
     hence x in dom g1 by A5,FUNCT_1:11;
    end;

    for y being Element of Y st y in dom g1 holds g1.y = max(-(g.y),0.)
    proof
     let y be Element of Y;
     assume y in dom g1; then
A9:  y in dom H & H.y in dom (max- f) by FUNCT_1:11;
     reconsider x = H.y as Element of X;
     g1.y = (max-f).x by FUNCT_1:13,A9; then
     g1.y = max(-(f.x),0.) by A9,MESFUNC2:def 3;
     hence g1.y = max(-(g.y),0.) by A2,A9,FUNCT_1:13;
    end;
    hence max-g = (max-f)*T" by A8,TARSKI:2,MESFUNC2:def 3;
end;
