let X be Subset of REAL; :: thesis: ( X is bounded_below implies X c= right_closed_halfline (lower_bound X) )
assume A1: X is bounded_below ; :: thesis: X c= right_closed_halfline (lower_bound X)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in right_closed_halfline (lower_bound X) )
assume A2: x in X ; :: thesis: x in right_closed_halfline (lower_bound X)
then reconsider x = x as Real ;
lower_bound X <= x by A1, A2, SEQ_4:def 2;
hence x in right_closed_halfline (lower_bound X) by XXREAL_1:236; :: thesis: verum