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