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
  for X being BCI-Algebra_with_Condition(S) holds for x,y,z being
  Element of X holds (x*z)\(y*z) <= x\y
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,y,z be Element of X;
  x <= y*(x\y) by Th12;
  then x*z <= (y*(x\y))*z by Th7;
  then x*z <= (y*z)*(x\y) by Th10;
  then (x*z)\(y*z) <= ((y*z)*(x\y))\(y*z) by BCIALG_1:5;
  then
A1: ((x*z)\(y*z)) \ (((y*z)*(x\y))\(y*z)) = 0.X;
  (((y*z)*(x\y))\(y*z)) <= x\y by Lm2;
  then (((y*z)*(x\y))\(y*z)) \ (x\y) = 0.X;
  then ((x*z)\(y*z)) \ (x\y) = 0.X by A1,BCIALG_1:3;
  hence thesis;
end;
