let X be bounded connected Subset of REAL ; :: thesis: ( not lower_bound X in X & upper_bound X in X implies X = ].(lower_bound X),(upper_bound X).] )
assume that
A1: not lower_bound X in X and
A2: upper_bound X in X ; :: thesis: X = ].(lower_bound X),(upper_bound X).]
thus X c= ].(lower_bound X),(upper_bound X).] by A1, Th26; :: according to XBOOLE_0:def 10 :: thesis: ].(lower_bound X),(upper_bound X).] c= X
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in ].(lower_bound X),(upper_bound X).] or x in X )
assume A3: x in ].(lower_bound X),(upper_bound X).] ; :: thesis: x in X
then reconsider x = x as Real ;
A4: x <= upper_bound X by A3, XXREAL_1:2;
lower_bound X < x by A3, XXREAL_1:2;
then (lower_bound X) - (lower_bound X) < x - (lower_bound X) by XREAL_1:16;
then consider r being real number such that
A5: r in X and
A6: r < (lower_bound X) + (x - (lower_bound X)) by A2, SEQ_4:def 5;
A7: [.r,(upper_bound X).] c= X by A2, A5, JCT_MISC:def 1;
x in [.r,(upper_bound X).] by A4, A6, XXREAL_1:1;
hence x in X by A7; :: thesis: verum