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 Th24:
  for X being BCI-Algebra_with_Condition(S) holds (0.X) |^ 2 = 0.X
proof
  let X be BCI-Algebra_with_Condition(S);
  0.X*0.X = (0.X*0.X) \ 0.X by BCIALG_1:2;
  then 0.X*0.X <= 0.X by Lm2;
  then
A1: (0.X*0.X) \ 0.X = 0.X;
  0.X <= 0.X*(0.X\0.X) by Th12;
  then 0.X <= 0.X*0.X by BCIALG_1:def 5;
  then 0.X \ (0.X*0.X) = 0.X;
  then 0.X*0.X = 0.X by A1,BCIALG_1:def 7;
  hence thesis by Th22;
end;
