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 Th4:
  for X being BCI-algebra holds X is p-Semisimple iff for x,y being
  Element of X st x\y = 0.X holds x=y
proof
  let X be BCI-algebra;
  thus X is p-Semisimple implies for x,y being Element of X st x\y = 0.X holds
  x=y
  proof
    assume
A1: X is p-Semisimple;
    for x,y being Element of X st x\y = 0.X holds x=y
    proof
      let x,y be Element of X;
      assume
A2:   x\y = 0.X;
      0.X\(x\y) = (y\x)\(0.X)` by A1,BCIALG_1:66
        .= (y\x)\0.X by BCIALG_1:def 5
        .= y\x by BCIALG_1:2;
      then y\x = 0.X by A2,BCIALG_1:def 5;
      hence thesis by A2,BCIALG_1:def 7;
    end;
    hence thesis;
  end;
  assume
A3: for x,y being Element of X st x\y = 0.X holds x=y;
  for x,y being Element of X holds x\(x\y) = y
  proof
    let x,y be Element of X;
    (x\(x\y))\y = (x\y)\(x\y) by BCIALG_1:7
      .= 0.X by BCIALG_1:def 5;
    hence thesis by A3;
  end;
  hence thesis;
end;
