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 Th16:
  for N being normal Subgroup of G holds
  A1 in Cosets N & A2 in Cosets N implies A1 * A2 in Cosets N
proof
  let N be normal Subgroup of G;
  assume that
A1: A1 in Cosets N and
A2: A2 in Cosets N;
  consider a such that
A3: A1 = a * N and
  A1 = N * a by A1,Th13;
  consider b such that
A4: A2 = b * N and
A5: A2 = N * b by A2,Th13;
  A1 * A2 = a * (N * (b * N)) by A3,A4,GROUP_3:10
    .= a * (b * N * N) by A4,A5,GROUP_3:13
    .= a * (b * (N * N)) by GROUP_4:45
    .= a * (b * N) by GROUP_2:76
    .= a * b * N by GROUP_2:105;
  hence thesis by GROUP_2:def 15;
end;
