reserve X,Y for set, p,x,x1,x2,y,y1,y2,z,z1,z2 for object;
reserve f,g,h for Function;

theorem
  g c= f & f is one-to-one implies rng g|`f = g
proof
  assume
A1: g c= f;
  assume
A2: f is one-to-one;
  for x,y being object holds [x,y] in (rng g|`f) implies [x,y] in g
  proof let x,y be object;
    assume
A3: [x,y] in (rng g|`f);
    then y in rng g by RELAT_1:def 12;
    then
A4: ex x1 being object st [x1,y] in g by XTUPLE_0:def 13;
    [x,y] in f by A3,RELAT_1:def 12;
    hence thesis by A1,A2,A4,Th9;
  end;
  then
A5: (rng g|`f) c= g;
  (rng g|`g) c= (rng g|`f) by A1,RELAT_1:101;
  then g c= (rng g|`f);
  hence thesis by A5;
end;
