reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem
  X c= Y implies X --> a c= Y --> a
proof
  assume
A2: X c= Y;
A3: now
    let x be object;
    assume
A4: x in dom(X --> a);
    then (X --> a).x = a by FUNCOP_1:7;
    hence (X --> a).x = (Y --> a).x by A2,A4,FUNCOP_1:7;
  end;
  dom(Y --> a) = Y;
  hence thesis by A2,A3,GRFUNC_1:2;
end;
