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

theorem Th54:
  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 Th24;
  then [.r,s.] c= [.p,q.] by A2;
  hence p <= r by A1,Th50;
  s in [.r,s.] by A1,Th1;
  hence thesis by A2,Th3;
end;
