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

theorem Th197:
  s < t implies ].r,t.[ \ ].s,t.[ = ].r,s.]
proof
  assume that
A1: s < t;
  let p;
  thus p in ].r,t.[ \ ].s,t.[ implies p in ].r,s.]
  proof
    assume
A2: p in ].r,t.[ \ ].s,t.[;
    then
A3: not p in ].s,t.[ by XBOOLE_0:def 5;
A4: r < p by A2,Th4;
    p <= s or t <= p by A3,Th4;
    hence thesis by A2,A4,Th2,Th4;
  end;
  assume
A5: p in ].r,s.];
  then
A6: p <= s by Th2;
A7: r < p by A5,Th2;
  p < t by A1,A6,XXREAL_0:2;
  then
A8: p in ].r,t.[ by A7,Th4;
  not p in ].s,t.[ by A6,Th4;
  hence thesis by A8,XBOOLE_0:def 5;
end;
