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 Th10:
  for N,N1,N2 be normal Subgroup of G,a,b be Element of G st
  the carrier of N = N1 * N2 holds (a * N1) * (b * N2) = (a * b) * N
proof
  let N,N1,N2 be normal Subgroup of G;
  let a,b be Element of G;
  assume
A1: the carrier of N = N1 * N2;
  (a * N1) * (b * N2) = a * N1 * b * N2 by GROUP_2:10
                     .= a * (N1 * b) * N2 by GROUP_2:29
                     .= a * (b * N1) * N2 by GROUP_3:117
                     .= (a * b) * N1 * N2 by GROUP_2:105
                     .= (a * b) * (N1 * N2) by GROUP_4:45;
  hence thesis by A1;
end;
