let X be real-bounded Subset of REAL; :: thesis: ( not lower_bound X in X & not upper_bound X in X implies X c= ].(lower_bound X),(upper_bound X).[ )
assume that
A1: not lower_bound X in X and
A2: not upper_bound X in X ; :: thesis: X c= ].(lower_bound X),(upper_bound X).[
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in ].(lower_bound X),(upper_bound X).[ )
assume A3: x in X ; :: thesis: x in ].(lower_bound X),(upper_bound X).[
then reconsider x = x as Real ;
x <= upper_bound X by A3, SEQ_4:def 1;
then A4: x < upper_bound X by A2, A3, XXREAL_0:1;
lower_bound X <= x by A3, SEQ_4:def 2;
then lower_bound X < x by A1, A3, XXREAL_0:1;
hence x in ].(lower_bound X),(upper_bound X).[ by A4, XXREAL_1:4; :: thesis: verum