reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th11:
  varcl (X \/ Y) = (varcl X) \/ (varcl Y)
proof
  set A = the set of all varcl a where a is Element of {X,Y};
  X \/ Y = union {X,Y} by ZFMISC_1:75;
  then
A1: varcl (X \/ Y) = union A by Th10;
  A = {varcl X, varcl Y}
  proof
    thus
    now
      let x be object;
      assume x in A;
      then consider a being Element of {X,Y} such that
A2:   x = varcl a;
      a = X or a = Y by TARSKI:def 2;
      hence x in {varcl X, varcl Y} by A2,TARSKI:def 2;
    end;
    let x be object;
    assume x in {varcl X, varcl Y};
    then x = varcl X & X in {X,Y} or x = varcl Y & Y in {X,Y} by TARSKI:def 2;
    hence thesis;
  end;
  hence thesis by A1,ZFMISC_1:75;
end;
