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

theorem Th9:
  Union (A1 (\/) A2) = Union A1 \/ Union A2
proof
  thus Union (A1 (\/) A2) c= Union A1 \/ Union A2
  proof
    let x be object;
    assume x in Union (A1 (\/) A2);
    then consider n1 being Nat such that
A1: x in (A1 (\/) A2).n1 by PROB_1:12;
A2: x in A1.n1 \/ A2.n1 by A1,Def2;
    per cases by A2,XBOOLE_0:def 3;
    suppose
      x in A1.n1;
      then x in Union A1 by PROB_1:12;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      x in A2.n1;
      then x in Union A2 by PROB_1:12;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  let x be object;
  assume
A3: x in Union A1 \/ Union A2;
  per cases by A3,XBOOLE_0:def 3;
  suppose
    x in Union A1;
    then consider n2 being Nat such that
A4: x in A1.n2 by PROB_1:12;
    x in A1.n2 \/ A2.n2 by A4,XBOOLE_0:def 3;
    then x in (A1 (\/) A2).n2 by Def2;
    hence thesis by PROB_1:12;
  end;
  suppose
    x in Union A2;
    then consider n3 being Nat such that
A5: x in A2.n3 by PROB_1:12;
    x in A1.n3 \/ A2.n3 by A5,XBOOLE_0:def 3;
    then x in (A1 (\/) A2).n3 by Def2;
    hence thesis by PROB_1:12;
  end;
end;
