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

theorem Th190:
  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,Th1;
    p < s or t < p by A3,Th1;
    hence thesis by A2,A4,Th1,Th3;
  end;
  assume
A5: p in [.r,s.[;
  then
A6: p < s by Th3;
A7: r <= p by A5,Th3;
  p <= t by A1,A6,XXREAL_0:2;
  then
A8: p in [.r,t.] by A7,Th1;
  not p in [.s,t.] by A6,Th1;
  hence thesis by A8,XBOOLE_0:def 5;
end;
