let X be interval Subset of REAL; :: thesis: ( not X is bounded_below & X is bounded_above & upper_bound X in X implies X = left_closed_halfline (upper_bound X) )
assume that
A1: not X is bounded_below and
A2: X is bounded_above and
A3: upper_bound X in X ; :: thesis: X = left_closed_halfline (upper_bound X)
thus X c= left_closed_halfline (upper_bound X) by A2, Th20; :: according to XBOOLE_0:def 10 :: thesis: left_closed_halfline (upper_bound X) c= X
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in left_closed_halfline (upper_bound X) or x in X )
assume A4: x in left_closed_halfline (upper_bound X) ; :: thesis: x in X
then reconsider x = x as Real ;
x is not LowerBound of X by A1;
then consider r being ExtReal such that
A5: r in X and
A6: x > r by XXREAL_2:def 2;
reconsider r = r as Real by A5;
x <= upper_bound X by A4, XXREAL_1:234;
then A7: x in [.r,(upper_bound X).] by A6, XXREAL_1:1;
[.r,(upper_bound X).] c= X by A3, A5, XXREAL_2:def 12;
hence x in X by A7; :: thesis: verum