reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th39:
  Union (A (\/) A1) = A \/ Union A1
proof
  thus Union (A (\/) A1) c= A \/ Union A1
  proof
    let x be object;
    assume
A1: x in Union (A (\/) A1);
A2: x in A or ex k st x in A1.k
    proof
      consider k such that
A3:   x in (A (\/) A1).k by A1,PROB_1:12;
      x in (A \/ A1.k) by A3,Def6;
      then x in A or x in A1.k by XBOOLE_0:def 3;
      hence thesis;
    end;
    per cases by A2;
    suppose
      x in A;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      ex k st x in A1.k;
      then x in Union A1 by PROB_1:12;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A4: x in A \/ Union A1;
  per cases by A4,XBOOLE_0:def 3;
  suppose
    x in A;
    then x in (A \/ A1.1) by XBOOLE_0:def 3;
    then x in (A (\/) A1).1 by Def6;
    hence thesis by PROB_1:12;
  end;
  suppose
    x in Union A1;
    then consider k such that
A5: x in A1.k by PROB_1:12;
    x in A \/ A1.k by A5,XBOOLE_0:def 3;
    then x in (A (\/) A1).k by Def6;
    hence thesis by PROB_1:12;
  end;
end;
