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 Th7: :: Isotonic Property
  for X being BCI-Algebra_with_Condition(S) holds for x,y,z being
  Element of X holds x <= y implies x*z <= y*z & z*x <= z*y
proof
  let X be BCI-Algebra_with_Condition(S);
  let x,y,z be Element of X;
  assume x <= y;
  then (x*z)\y <= (x*z)\x by BCIALG_1:5;
  then
A1: ((x*z)\y)\((x*z)\x) = 0.X;
  (x*z)\x <= z by Lm2;
  then ((x*z)\x)\z = 0.X;
  then ((x*z)\y)\z = 0.X by A1,BCIALG_1:3;
  then
A2: (x*z)\y <= z;
  x*z = z*x & y*z = z*y by Th6;
  hence thesis by A2,Lm2;
end;
