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

theorem Th189:
  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,Th4;
    hence thesis by A2,A4,Th3,Th4;
  end;
  assume
A5: p in [.s,t.[;
  then
A6: s <= p by Th3;
  then
A7: r < p by A1,XXREAL_0:2;
  p < t by A5,Th3;
  then
A8: p in ].r,t.[ by A7,Th4;
  not p in ].r,s.[ by A6,Th4;
  hence thesis by A8,XBOOLE_0:def 5;
end;
