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,y,z holds (x\y)\(z\y)=x\z
proof
  thus X is p-Semisimple implies for x,y,z holds (x\y)\(z\y)=x\z by Lm11;
  assume
A1: for x,y,z holds (x\y)\(z\y)=x\z;
  for x,z holds z`\x` = x\z
  proof
    let x,z;
    (z\x)`=(x\x)\(z\x)by Def5;
    then (z\x)`=x\z by A1;
    hence thesis by Th9;
  end;
  hence thesis by Th59;
end;
