reserve P,Q,X,Y,Z for set, p,x,x9,x1,x2,y,z for object;
reserve D for non empty set;
reserve A,B for non empty set;
reserve Y for non empty set,
  f for Function of X,Y,
  p for PartFunc of Y,Z,
  x for Element of X;
reserve g for Function of X,X;
reserve X,Y for non empty set,
  Z,S,T for set,
  f for Function of X,Y,
  g for PartFunc of Y,Z,
  x for Element of X;

theorem
  rng f c= dom (g|S) & S c= T implies (g|S)/*f = (g|T)/*f
proof
  assume
A1: rng f c= dom (g|S);
  assume S c= T;
  then g|S c= g|T by RELAT_1:75;
  then
A2: dom (g|S) c= dom (g|T) by RELAT_1:11;
  thus (g|S)/*f = g/*f by A1,Th116
    .= (g|T)/*f by A1,A2,Th116,XBOOLE_1:1;
end;
