reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem
  X is p-Semisimple iff for x being Element of X holds x is atom
proof
  thus X is p-Semisimple implies for x being Element of X holds x is atom
  proof
    assume
A1: X is p-Semisimple;
    let x be Element of X;
    now
      let z be Element of X;
      assume z\x=0.X;
      then z\0.X = x by A1;
      hence z=x by Th2;
    end;
    hence thesis;
  end;
  assume
A2: for x being Element of X holds x is atom;
  for x,y being Element of X holds x\(x\y) = y
  proof
    let x,y be Element of X;
    y is atom & (x\(x\y))\y=0.X by A2,Th1;
    hence thesis;
  end;
  hence thesis;
end;
