let X be Subset of REAL; :: thesis: ( X is bounded_above & not upper_bound X in X implies X c= left_open_halfline (upper_bound X) )
assume that
A1: X is bounded_above and
A2: not upper_bound X in X ; :: thesis: X c= left_open_halfline (upper_bound X)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in X or x in left_open_halfline (upper_bound X) )
assume A3: x in X ; :: thesis: x in left_open_halfline (upper_bound X)
then reconsider x = x as Real ;
x <= upper_bound X by A1, A3, SEQ_4:def 1;
then x < upper_bound X by A2, A3, XXREAL_0:1;
hence x in left_open_halfline (upper_bound X) by XXREAL_1:233; :: thesis: verum