let x be set ; :: thesis: for p, q being ext-real number holds
( not x in [.p,q.] or x in ].p,q.[ or x = p or x = q )
let p, q be ext-real number ; :: thesis: ( not x in [.p,q.] or x in ].p,q.[ or x = p or x = q )
assume A1: x in [.p,q.] ; :: thesis: ( x in ].p,q.[ or x = p or x = q )
then reconsider s = x as ext-real number ;
( p <= s & s <= q ) by A1, Th1;
then ( ( p = s or p < s ) & ( s = q or s < q ) ) by XXREAL_0:1;
hence ( x in ].p,q.[ or x = p or x = q ) by Th4; :: thesis: verum