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 Th9:
  B /\ N is normal Subgroup of B & N /\ B is normal Subgroup of B
proof
  thus B /\ N is normal Subgroup of B
  proof
    reconsider A = B /\ N as Subgroup of B by GROUP_2:88;
    now
      let b be Element of B;
      thus b * A c= A * b
      proof
        let x be object;
        assume x in b * A;
        then consider a being Element of B such that
A1:     x = b * a and
A2:     a in A by GROUP_2:103;
        reconsider a9 = a, b9 = b as Element of G by GROUP_2:42;
        a in N & x = b9 * a9 by A1,A2,GROUP_2:43,82;
        then
A3:     x in b9 * N by GROUP_2:103;
        reconsider x9 = x as Element of B by A1;
A4:     b9" = b" by GROUP_2:48;
        b9 * N c= N * b9 by GROUP_3:118;
        then consider b1 such that
A5:     x = b1 * b9 and
A6:     b1 in N by A3,GROUP_2:104;
        reconsider x99 = x as Element of G by A5;
        b1 = x99 * b9" by A5,GROUP_1:14;
        then
A7:     b1 = x9 * b" by A4,GROUP_2:43;
        then reconsider b19 = b1 as Element of B;
        b1 in B by A7;
        then
A8:     b19 in A by A6,GROUP_2:82;
        b19 * b = x by A5,GROUP_2:43;
        hence thesis by A8,GROUP_2:104;
      end;
    end;
    hence thesis by GROUP_3:118;
  end;
  hence thesis;
end;
