reserve G for Group;
reserve A,B for non empty Subset of G;
reserve N,H,H1,H2 for Subgroup of G;
reserve x,a,b for Element of G;

theorem
  N ` A \/ N ` B c= N ` (A \/ B)
proof
  let x be object;
  assume
A1: x in N ` A \/  N ` B;
  then reconsider x as Element of G;
  per cases by A1,XBOOLE_0:def 3;
  suppose x in N ` A;
    then x * N c= A \/ B by Th12,XBOOLE_1:10;
    hence thesis;
  end;
  suppose x in N ` B;
    then x * N c= A \/ B by Th12,XBOOLE_1:10;
    hence thesis;
  end;
end;
