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 Th59:
  X is p-Semisimple iff for x,z holds z`\x` = x\z
proof
  thus X is p-Semisimple implies for x,z holds z`\x` = x\z by Lm10;
  assume
A1: for x,z holds z`\x` = x\z;
  for x holds x`` = x
  proof
    let x;
    (0.X)`\x` = x\0.X by A1;
    then (x`)` = x\0.X by Th2;
    hence thesis by Th2;
  end;
  hence thesis by Th54;
end;
