let x be set ; :: thesis: for p, q being ext-real number st x in [.p,q.[ holds
( x in [.p,q.] & x <> q )
let p, q be ext-real number ; :: thesis: ( x in [.p,q.[ implies ( x in [.p,q.] & x <> q ) )
assume A1: x in [.p,q.[ ; :: thesis: ( x in [.p,q.] & x <> q )
then reconsider s = x as ext-real number ;
A2: p <= s by A1, Th3;
s < q by A1, Th3;
hence ( x in [.p,q.] & x <> q ) by A2, Th1; :: thesis: verum