reserve G for strict Group,
  a,b,x,y,z for Element of G,
  H,K for strict Subgroup of G,
  p for Element of NAT,
  A for Subset of G;

theorem Th17:
  z in H * x * K iff ex g,h being Element of G st
  z = g * x * h & g in H & h in K
proof
  thus z in H * x * K implies ex g,h being Element of G st
  z = g * x * h & g in H & h in K
  proof
    assume z in H * x * K;
    then consider g1, g2 being Element of G such that
A1: z = g1 * g2 and
A2: g1 in H * x and
A3: g2 in K by GROUP_2:94;
    ex h1 being Element of G st ( g1 = h1 * x)&( h1 in H) by A2,GROUP_2:104;
    hence thesis by A1,A3;
  end;
  given g,h being Element of G such that
A4: z = g * x * h and
A5: g in H and
A6: h in K;
  ex g1, g2 being Element of G st z = g1 * g2 & g1 in H * x & g2 in K
    by A5,A4,A6,GROUP_2:104;
  hence thesis by GROUP_2:94;
end;
