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;
reserve N1,N2 for Subgroup of G;

theorem
  for N1,N2 be strict normal Subgroup of G
  ex N being strict normal Subgroup of G st
    the carrier of N = N1 * N2 & N1 ` H * N2 ` H c= N ` H
proof
  let N1,N2 be strict normal Subgroup of G;
  consider N be strict normal Subgroup of G such that
A1:the carrier of N = N1 * N2 by Th8;
  N1 ` H * N2 ` H c= N ` H
  proof
    let x be object;
    assume
A2: x in N1 ` H * N2 ` H;
    then reconsider x as Element of G;
    consider a,b be Element of G such that
A3: x = a * b & a in N1 ` H & b in N2 ` H by A2;
    a * N1 c= carr(H) & b * N2 c= carr(H) by A3,Th49;
    then (a * N1) * (b * N2) c= carr(H) * carr(H) by GROUP_3:4; then
A4: (a * N1) * (b * N2) c= carr(H) by GROUP_2:76;
    (a * N1) * (b * N2) = (a * b) * N by A1,Th10;
    hence thesis by A3,A4;
  end;
  hence thesis by A1;
end;
