
theorem Th23:
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
  f is nonnegative iff g is nonnegative
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
A1: T is bijective & g = f*T";

A2: dom T = X by FUNCT_2:def 1;
A3: dom f c= X;
    g*T = f*(T"*T) by A1,RELAT_1:36; then
    g*T = f* (id dom T) by FUNCT_1:39,A1; then
A4: g*T = f by RELAT_1:51,A2,A3;
    hereby assume f is nonnegative; then
A5:  rng f is nonnegative;
     now let y be ExtReal;
      assume y in rng g; then
      consider x be object such that
A6:   x in dom g & y = g.x by FUNCT_1:def 3;
A7:   y = f.(T".x) by A1,A6,FUNCT_1:12;
      x in dom (T") & T".x in dom f by A1,A6,FUNCT_1:11;
      hence 0. <= y by A5,A7,FUNCT_1:3;
     end; then
     rng g is nonnegative;
     hence g is nonnegative;
    end;

    assume
A8: g is nonnegative;
    now let y be ExtReal;
     assume y in rng f; then
     consider x be object such that
A9:  x in dom (g*T) & y = (g*T).x by A4,FUNCT_1:def 3;
A10: x in dom T & T.x in dom g by A9,FUNCT_1:11;
     y = g.(T.x) by FUNCT_1:12,A9;
     hence 0. <= y by A8,A10,FUNCT_1:3,SUPINF_2:def 9;
    end; then
    rng f is nonnegative;
    hence f is nonnegative;
end;
