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 Th62: ::TL2232:
  X is p-Semisimple iff for x st x`=0.X holds x=0.X
proof
  thus X is p-Semisimple implies for x st x`=0.X holds x=0.X
  proof
    assume
A1: X is p-Semisimple;
    let x;
    assume x`=0.X;
    then (x`)`=0.X by Def5;
    hence thesis by A1;
  end;
  assume
A2: for x st x`=0.X holds x=0.X;
  for x holds x`` = x
  proof
    let x;
    (x\x``)`=x`\x``` by Th9
      .=x`\x` by Th8
      .=0.X by Def5;
    then
A3: x\((x)`)`=0.X by A2;
    ((x)`)`\x=0.X by Th1;
    hence thesis by A3,Def7;
  end;
  hence thesis by Th54;
end;
