reserve n for Element of NAT;
reserve i for Integer;
reserve G,H,I for Group;
reserve A,B for Subgroup of G;
reserve N for normal Subgroup of G;
reserve a,a1,a2,a3,b,b1 for Element of G;
reserve c,d for Element of H;
reserve f for Function of the carrier of G, the carrier of H;
reserve x,y,y1,y2,z for set;
reserve A1,A2 for Subset of G;

theorem Th8:
  for N being normal Subgroup of G holds N is Subgroup of B implies
  N is normal Subgroup of B
proof
  let N be normal Subgroup of G;
  assume N is Subgroup of B;
  then reconsider N9 = N as Subgroup of B;
  now
    let n be Element of B;
    thus n * N9 c= N9 * n
    proof
      let x be object;
      assume x in n * N9;
      then consider a being Element of B such that
A1:   x = n * a and
A2:   a in N9 by GROUP_2:103;
      reconsider a9 = a, n9 = n as Element of G by GROUP_2:42;
      x = n9 * a9 by A1,GROUP_2:43;
      then
A3:   x in n9 * N by A2,GROUP_2:103;
      n9 * N c= N * n9 by GROUP_3:118;
      then consider a1 such that
A4:   x = a1 * n9 and
A5:   a1 in N by A3,GROUP_2:104;
      a1 in N9 by A5;
      then a1 in B by GROUP_2:40;
      then reconsider a19 = a1 as Element of B;
      x = a19 * n by A4,GROUP_2:43;
      hence thesis by A5,GROUP_2:104;
    end;
  end;
  hence thesis by GROUP_3:118;
end;
