 reserve m,n for Nat;
 reserve i,j for Integer;
 reserve S for non empty multMagma;
 reserve r,r1,r2,s,s1,s2,t for Element of S;
 reserve G for Group-like non empty multMagma;
 reserve e,h for Element of G;
 reserve G for Group;
 reserve f,g,h for Element of G;

theorem Th17:
  g * h = h * g iff (g * h)" = g" * h"
proof
  thus g * h = h * g implies (g * h)" = g" * h" by Th16;
  assume (g * h)" = g" * h";
  then
A1: (h * g) * (g * h)" = h * g * g" * h" by Def3
    .= h * (g * g") * h" by Def3
    .= h * 1_G * h" by Def5
    .= h * h" by Def4
    .= 1_G by Def5;
  (g * h) * (g * h)" = 1_G by Def5;
  hence thesis by A1,Th6;
end;
