
theorem Th5:
  for X1,Y1, X2,Y2 being non empty set for f being Function of X1,
  X2, g being Function of Y1,Y2 st f c= g holds f* c= g*
proof
  let X1,Y1, X2,Y2 be non empty set;
  let f be Function of X1,X2, g be Function of Y1,Y2;
A1: dom g = Y1 by FUNCT_2:def 1;
  assume
A2: f c= g;
A3: dom f = X1 by FUNCT_2:def 1;
  then
A4: X1* c= Y1* by A1,A2,FINSEQ_1:62,RELAT_1:11;
A5: now
    let x be object;
    assume x in X1*;
    then reconsider p = x as Element of X1*;
A6: (f*).p = f*p by LANG1:def 13;
    (rng p) /\ Y1 c= X1;
    then
A7: f*p = g*p by A3,A1,A2,Th2;
    p in X1*;
    hence (f*).x = (g*).x by A4,A6,A7,LANG1:def 13;
  end;
A8: dom (g*) = Y1* by FUNCT_2:def 1;
  dom (f*) = X1* by FUNCT_2:def 1;
  hence thesis by A8,A4,A5,GRFUNC_1:2;
end;
