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 \/ B) = N ~ A \/ N ~ B
proof
  thus N ~ (A \/ B) c= N ~ A \/ N ~ B
  proof
    let x be object;
    assume
A1: x in N ~ (A \/ B);
    then reconsider x as Element of G;
    x * N meets (A \/ B) by A1,Th14;
    then x * N meets A or x * N meets B by XBOOLE_1:70;
    then x in N ~ A or x in N ~ B;
    hence thesis by XBOOLE_0:def 3;
  end;
  let x be object;
  assume
A2:x in N ~ A \/ N ~ B;
  then reconsider x as Element of G;
  x in N ~ A or x in N ~ B by A2,XBOOLE_0:def 3;
  then
  x * N meets A or x * N meets B by Th14;
  then x * N meets (A \/ B) by XBOOLE_1:70;
  hence thesis;
end;
