let X be connected Subset of REAL ; :: thesis: ( not X is bounded_above & not X is bounded_below implies X = REAL )
assume that
A1: not X is bounded_above and
A2: not X is bounded_below ; :: thesis: X = REAL
thus X c= REAL ; :: according to XBOOLE_0:def 10 :: thesis: REAL c= X
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in REAL or x in X )
assume x in REAL ; :: thesis: x in X
then reconsider x = x as Real ;
consider r being real number such that
A3: ( r in X & r > x ) by A1, SEQ_4:def 1;
consider s being real number such that
A4: ( s in X & s < x ) by A2, SEQ_4:def 2;
( [.s,r.] c= X & x in [.s,r.] ) by A3, A4, JCT_MISC:def 1, XXREAL_1:1;
hence x in X ; :: thesis: verum