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;
    consider x1 being Element of G such that
A2: x1 = x & x1 * N c= A /\ B by A1;
    x * N c= A & x * N c= B by A2,XBOOLE_1:18;
    then x in N ` A & x in N ` B;
    hence thesis by XBOOLE_0:def 4;
  end;
  let x be object;
  assume
A3: x in N ` A /\ N ` B;
  then reconsider x as Element of G;
  x in N ` A & x in N ` B by A3,XBOOLE_0:def 4;
  then x * N c= A & x * N c= B by Th12;
  then x * N c= A /\ B by XBOOLE_1:19;
  hence thesis;
end;
