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 y\x=z\x holds y=z
proof
  thus X is p-Semisimple implies for x,y,z st y\x=z\x holds y=z by Lm14;
  assume
A1: for x,y,z st y\x=z\x holds y=z;
  for x,y holds x\(x\y) = y
  proof
    let x,y;
    (x\(x\y))\y = 0.X by Th1;
    then (x\(x\y))\y = y\y by Def5;
    hence thesis by A1;
  end;
  hence thesis;
end;
