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 in dom f implies x .--> f.x c= f
proof
A1: now
    let y be object;
    assume y in dom (x .--> f.x);
    then x = y by FUNCOP_1:75;
    hence (x .--> f.x).y = f.y by FUNCOP_1:72;
  end;
  assume
A2: x in dom f;
  dom (x .--> f.x) c= dom f
  by A2,FUNCOP_1:75;
  hence thesis by A1,GRFUNC_1:2;
end;
