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 st x\y=x\z holds y=z
proof
  thus X is p-Semisimple implies for x,y,z st x\y=x\z holds y=z by Lm12;
  assume
A1: for x,y,z st x\y=x\z holds y=z;
  for x st x`=0.X holds x=0.X
  proof
    let x;
    assume x`=0.X;
    then x`=(0.X)` by Def5;
    hence thesis by A1;
  end;
  hence thesis by Th62;
end;
