reserve X for BCK-algebra;
reserve x,y for Element of X;
reserve IT for non empty Subset of X;

theorem
  for X being BCI-algebra holds (X is p-Semisimple implies X is
  BCI-commutative & X is BCI-weakly-commutative )
proof
  let X being BCI-algebra;
  assume
A1: X is p-Semisimple;
A2: for x,y being Element of X holds (x\(x\y))\(0.X\(x\y)) = y\(y\x)
  proof
    let x,y be Element of X;
    (0.X\(x\y)) = ((0.X\0.X)\(x\y)) by BCIALG_1:def 5
      .= (0.X\x)\(0.X\y) by A1,BCIALG_1:64
      .= (y\x)\(0.X\0.X) by A1,BCIALG_1:58
      .= (y\x)\0.X by BCIALG_1:def 5
      .= y\x by BCIALG_1:2;
    hence thesis by A1;
  end;
  for x,y being Element of X st x\y=0.X holds x = y\(y\x) by A1;
  hence thesis by A2;
end;
