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

proof

let x be Real; :: according to MEMBERED:def 9 :: thesis: ( not x in ].-infty,s.[ or x in { g where g is Real : g < s } )
assume A1: x in ].-infty,s.[ ; :: thesis: x in { g where g is Real : g < s }
A2: x < s by A1, Th4;
thus x in { g where g is Real : g < s } by A2; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { g where g is Real : g < s } or x in ].-infty,s.[ )
assume x in { g where g is Real : g < s } ; :: thesis: x in ].-infty,s.[
then consider g being Real such that
A3: x = g and
A4: g < s ;
g in REAL by XREAL_0:def 1;
then -infty < g by XXREAL_0:12;
hence x in ].-infty,s.[ by A3, A4, Th4; :: thesis: verum