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 being Element of
  X holds (x\0.X)*(0.X\x) = x
proof
  let X be BCK-Algebra_with_Condition(S);
  for x being Element of X holds (x\0.X)*(0.X\x) = x
  proof
    let x be Element of X;
A1: (0.X\x) = x` .= 0.X by BCIALG_1:def 8;
    (x\0.X) = x by BCIALG_1:2;
    hence thesis by A1,Th8;
  end;
  hence thesis;
end;
