let s be ext-real number ; :: thesis: ].s,+infty .[ = { g where g is Real : s < g }
thus ].s,+infty .[ c= { g where g is Real : s < g } :: according to XBOOLE_0:def 10 :: thesis: { g where g is Real : s < g } c= ].s,+infty .[

proof

let x be real number ; :: according to MEMBERED:def 9 :: thesis: ( not x in ].s,+infty .[ or x in { g where g is Real : s < g } )
assume x in ].s,+infty .[ ; :: thesis: x in { g where g is Real : s < g }
then B1: ( s < x & x < +infty ) by Th4;
then x in REAL by XXREAL_0:48;
hence x in { g where g is Real : s < g } by B1; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in { g where g is Real : s < g } or x in ].s,+infty .[ )
assume x in { g where g is Real : s < g } ; :: thesis: x in ].s,+infty .[
then consider g being Real such that
A2: ( x = g & s < g ) ;
g < +infty by XXREAL_0:9;
hence x in ].s,+infty .[ by A2, Th4; :: thesis: verum