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