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

theorem Th21:
  g c= f implies f|dom g = g
proof
  assume
A1: g c= f;
  for x,y being object holds [x,y] in (f|dom g) implies [x,y] in g
  proof let x,y be object;
    assume
A2: [x,y] in (f|dom g);
    then x in dom g by RELAT_1:def 11;
    then
A3: [x,g.x] in g by FUNCT_1:def 2;
    [x,y] in f by A2,RELAT_1:def 11;
    hence thesis by A1,A3,FUNCT_1:def 1;
  end;
  then
A4: (f|dom g) c= g;
  (g|dom g) c= (f|dom g) by A1,RELAT_1:76;
  then g c= (f|dom g);
  hence thesis by A4;
end;
