theorem Th28:
  p is_less_than X iff p% is_<=_than X
proof
  thus p is_less_than X implies p% is_<=_than X
  proof
    assume
A1: for q st q in X holds p [= q;
    let p9;
A2: (%p9)% = %p9;
    assume p9 in X;
    then p [= %p9 by A1;
    hence thesis by A2,Th7;
  end;
  assume
A3: for q9 st q9 in X holds p% <= q9;
  let q;
  assume q in X;
  then p% <= q% by A3;
  hence thesis by Th7;
end;
