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 Th54:
  X is p-Semisimple iff for x being Element of X holds x`` = x
proof
  (for x being Element of X holds x`` = x) implies X is p-Semisimple
  proof
    assume
A1: for x being Element of X holds x`` = x;
    now
      let x,y be Element of X;
A2:   (x\(x\y))\y=0.X by Th1;
      y\(x\(x\y))=(y\(x\(x\y)))`` by A1
        .=(y`\(x\(x\y))`)` by Th9
        .=0.X \(((0.X \y)\((x\(x\y))`))\0.X) by Th2
        .=0.X \(((0.X \y)\((x\(x\y))`))\((x\(x\y))\y)) by Th1
        .=0.X \0.X by Th1
        .=0.X by Def5;
      hence x\(x\y) = y by A2,Def7;
    end;
    hence thesis;
  end;
  hence thesis;
end;
