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 BCK-Algebra_with_Condition(S) holds for x,y being Element
  of X holds (x\y)*(y\x) <= x*y
proof
  let X be BCK-Algebra_with_Condition(S);
  for x,y being Element of X holds (x\y)*(y\x) <= x*y
  proof
    let x,y be Element of X;
    ((x\y)*(y\x))\(x\y) <= (y\x) by Lm2;
    then
A1: (((x\y)*(y\x))\(x\y))\y <= (y\x)\y by BCIALG_1:5;
    (y\x)\y = (y\y)\x by BCIALG_1:7
      .= x` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    then ((x\y)*(y\x))\((x\y)*y) <= 0.X by A1,Th11;
    then ((x\y)*(y\x))\((x\y)*y) \ 0.X = 0.X;
    then
A2: ((x\y)*(y\x))\((x\y)*y) = 0.X by BCIALG_1:2;
    (y*(x\y))\y <= x\y by Lm2;
    then
A3: ((y*(x\y))\y)\x <= (x\y)\x by BCIALG_1:5;
    (x\y)\x = (x\x)\y by BCIALG_1:7
      .= y` by BCIALG_1:def 5
      .= 0.X by BCIALG_1:def 8;
    then (y*(x\y))\(y*x) <= 0.X by A3,Th11;
    then ((y*(x\y))\(y*x)) \ 0.X = 0.X;
    then
A4: (y*(x\y))\(y*x) = 0.X by BCIALG_1:2;
    (x\y)*y = y*(x\y) by Th6;
    then ((x\y)*(y\x))\(y*x) = 0.X by A2,A4,BCIALG_1:3;
    then (x\y)*(y\x) <= (y*x);
    hence thesis by Th6;
  end;
  hence thesis;
end;
