theorem Th30:
  X is_less_than p iff X is_<=_than p%
proof
  thus X is_less_than p implies X is_<=_than p%
  proof
    assume
A1: for q st q in X holds q [= p;
    let p9;
A2: (%p9)% = %p9;
    assume p9 in X;
    then %p9 [= p by A1;
    hence thesis by A2,Th7;
  end;
  assume
A3: for q9 st q9 in X holds q9 <= p%;
  let q;
  assume q in X;
  then q% <= p% by A3;
  hence thesis by Th7;
end;
