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 Th8: :: Unit Element
  for X being BCI-Algebra_with_Condition(S) holds for x being
  Element of X holds 0.X*x = x & x*0.X = x
proof
  let X be BCI-Algebra_with_Condition(S);
  let x be Element of X;
  (x\0.X)\x = (x\x)\0.X by BCIALG_1:7
    .= 0.X\0.X by BCIALG_1:def 5
    .= 0.X by BCIALG_1:def 5;
  then x\0.X <= x;
  then x <= 0.X*x by Lm2;
  then
A1: x\(0.X*x) = 0.X;
  (0.X*x)\0.X <= x by Lm2;
  then (0.X*x) <= x by BCIALG_1:2;
  then (0.X*x)\x = 0.X;
  then 0.X*x = x by A1,BCIALG_1:def 7;
  hence thesis by Th6;
end;
