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

theorem Th182:
  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,Th1;
    p < r or s < p by A3,Th1;
    hence thesis by A2,A4,Th1,Th2;
  end;
  assume
A5: p in ].s,t.];
  then
A6: s < p by Th2;
  then
A7: r <= p by A1,XXREAL_0:2;
  p <= t by A5,Th2;
  then
A8: p in [.r,t.] by A7,Th1;
  not p in [.r,s.] by A6,Th1;
  hence thesis by A8,XBOOLE_0:def 5;
end;
