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,u holds (x\y)\(z\u)=(x\z)\(y\u)
proof
  thus X is p-Semisimple implies for x,y,z,u holds (x\y)\(z\u)=(x\z)\(y\u) by
Lm9;
  assume
A1: for x,y,z,u holds (x\y)\(z\u)=(x\z)\(y\u);
  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)\(x\x) by A1;
    then (z\x)`=(x\z)\0.X by Def5;
    then (z\x)`=x\z by Th2;
    hence thesis by Th9;
  end;
  hence thesis by Th59;
end;
