reserve x for set,
  p,q,r,s,t,u for ExtReal,
  g for Real,
  a for Element of ExtREAL;

theorem Th58:
  r <= s & [.r,s.] c= ].p,q.] implies p < r & s <= q
proof
  assume that
A1: r <= s and
A2: [.r,s.] c= ].p,q.];
  ].p,q.] c= [.p,q.] by Th23;
  then
A3: [.r,s.] c= [.p,q.] by A2;
  r in [.r,s.] by A1,Th1;
  hence p < r by A2,Th2;
  thus thesis by A1,A3,Th50;
end;
