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

theorem Th301:
  r < s & p <= q implies ].r,q.[ \ [.p,s.[ = ].r,p.[ \/ [.s,q.[
proof
  assume that
A1: r < s and
A2: p <= q;
  let x be ExtReal;
  thus x in ].r,q.[ \ [.p,s.[ implies x in ].r,p.[ \/ [.s,q.[
  proof
    assume
A3: x in ].r,q.[ \ [.p,s.[;
    then
A4: not x in [.p,s.[ by XBOOLE_0:def 5;
A5: r < x by A3,Th4;
A6: x < q by A3,Th4;
    not(p <= x & x < s) by A4,Th3;
    then x in ].r,p.[ or x in [.s,q.[ by A5,A6,Th3,Th4;
    hence thesis by XBOOLE_0:def 3;
  end;
  assume x in ].r,p.[ \/ [.s,q.[;
  then x in ].r,p.[ or x in [.s,q.[ by XBOOLE_0:def 3;
  then
A7: r < x & x < p or s <= x & x < q by Th3,Th4;
  then
A8: r < x by A1,XXREAL_0:2;
  x < q by A2,A7,XXREAL_0:2;
  then
A9: x in ].r,q.[ by A8,Th4;
  not x in [.p,s.[ by A7,Th3;
  hence thesis by A9,XBOOLE_0:def 5;
end;
