 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 Th18:
  g * h = h * g iff g" * h" = h" * g"
proof
  thus g * h = h * g implies g" * h" = h" * g"
  proof
    assume
A1: g * h = h * g;
    hence g" * h" = (g * h)" by Th16
      .= h" * g" by A1,Th17;
  end;
  assume
A2: g" * h" = h" * g";
  thus g * h = (g * h)"" .= (h" * g")" by Th16
    .= h"" * g"" by A2,Th16
    .= h * g;
end;
