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 Th8:
  for G being Group, N1,N2 being strict normal Subgroup of G
  ex M being strict normal Subgroup of G st the carrier of M = N1 * N2
proof
  let G be Group, N1,N2 be strict normal Subgroup of G;
  consider M being strict Subgroup of G such that
A1:the carrier of M = N1 * N2 by Th7;
  for x,y be Element of G st y in M holds x * y * x" in M
  proof
    let x,y be Element of G;
    assume y in M;
    then y in the carrier of M by STRUCT_0:def 5;
    then consider a,b be Element of G such that
A2: y = a * b & a in N1 & b in N2 by A1,Th6;
A3: x * y * x" =((x * a) * b) * x" by A2,GROUP_1:def 3
              .=(x * a) * (b * x") by GROUP_1:def 3
              .=(x * a) * 1_G * (b * x") by GROUP_1:def 4
              .=(x * a) * (x" * x) * (b * x") by GROUP_1:def 5
              .=(x * a) * x" * x * (b * x") by GROUP_1:def 3
              .=((x * a) * x") * (x * (b * x")) by GROUP_1:def 3
              .=(x * a * x") * (x * b * x") by GROUP_1:def 3;
    x * a * x" in N1 & x * b * x" in N2 by A2,Th4;
    then x * y * x" in N1 * N2 by A3,Th6;
    hence thesis by A1,STRUCT_0:def 5;
  end;
  then M is normal Subgroup of G by Th5;
  hence thesis by A1;
end;
