reserve X for non empty BCIStr_1;
reserve d for Element of X;
reserve n,m,k for Nat;
reserve f for sequence of  the carrier of X;

theorem Th6: :: Commutativity
  for X being BCI-Algebra_with_Condition(S) holds for x,y being
  Element of X holds x*y = y*x
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,y be Element of X;
  (y*x)\y <= x by Lm2;
  then ((y*x)\y)\x = 0.X;
  then ((y*x)\x)\y = 0.X by BCIALG_1:7;
  then (y*x)\x <= y;
  then (y*x) <= (x*y) by Lm2;
  then
A1: (y*x)\(x*y) = 0.X;
  (x*y)\x <= y by Lm2;
  then ((x*y)\x)\y = 0.X;
  then ((x*y)\y)\x = 0.X by BCIALG_1:7;
  then (x*y)\y <= x;
  then (x*y) <= (y*x) by Lm2;
  then (x*y)\(y*x) = 0.X;
  hence thesis by A1,BCIALG_1:def 7;
end;
