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

theorem Th187:
  r < s implies ].r,t.[ \ ].r,s.] = ].s,t.[
proof
  assume that
A1: r < s;
  let p;
  thus p in ].r,t.[ \ ].r,s.] implies p in ].s,t.[
  proof
    assume
A2: p in ].r,t.[ \ ].r,s.];
    then
A3: not p in ].r,s.] by XBOOLE_0:def 5;
A4: p < t by A2,Th4;
    p <= r or s < p by A3,Th2;
    hence thesis by A2,A4,Th4;
  end;
  assume
A5: p in ].s,t.[;
  then
A6: s < p by Th4;
  then
A7: r < p by A1,XXREAL_0:2;
  p < t by A5,Th4;
  then
A8: p in ].r,t.[ by A7,Th4;
  not p in ].r,s.] by A6,Th2;
  hence thesis by A8,XBOOLE_0:def 5;
end;
