let X be non empty bounded connected Subset of REAL ; :: thesis: ( not lower_bound X in X & not upper_bound X in X implies X = ].(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 = ].(lower_bound X),(upper_bound X).[
thus X c= ].(lower_bound X),(upper_bound X).[ by A1, A2, Th30; :: 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 ;
lower_bound X < x by A3, XXREAL_1:4;
then (lower_bound X) - (lower_bound X) < x - (lower_bound X) by XREAL_1:16;
then consider s being real number such that
A4: s in X and
A5: s < (lower_bound X) + (x - (lower_bound X)) by SEQ_4:def 5;
x < upper_bound X by A3, XXREAL_1:4;
then x - x < (upper_bound X) - x by XREAL_1:16;
then consider r being real number such that
A6: r in X and
A7: (upper_bound X) - ((upper_bound X) - x) < r by SEQ_4:def 4;
A8: [.s,r.] c= X by A4, A6, JCT_MISC:def 1;
x in [.s,r.] by A5, A7, XXREAL_1:1;
hence x in X by A8; :: thesis: verum