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;

theorem
  for T,S being set, f being Function of T,S, A,B being Subset-Family of T
  st A c= B holds f.:A c= f.:B
proof
  let T,S be set;
  let f be Function of T,S;
  let A,B be Subset-Family of T;
  assume
A1: A c= B;
  let x be object;
  assume
A2: x in f.:A;
  then reconsider x as Subset of S;
  ex C being Subset of T st C in B & x = f.:C
  proof
    consider C being Subset of T such that
A3: C in A & x = f.:C by A2,Def10;
    take C;
    thus thesis by A1,A3;
  end;
  hence thesis by Def10;
end;
