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

theorem
  p<r & p<s & r<q & s<q implies ].p,r.] \/ ].r,s.[ \/ [.s,q.[ = ].p,q.[
proof
  assume that
A1: p < r and
A2: p < s and
A3: r < q and
A4: s < q;
  per cases;
  suppose r < s;
    hence ].p,r.] \/ ].r,s.[ \/ [.s,q.[ = ].p,s.[ \/ [.s,q.[ by A1,Th171
      .= ].p,q.[ by A2,A4,Th173;
  end;
  suppose
A5: s <= r;
    hence ].p,r.] \/ ].r,s.[ \/ [.s,q.[ = ].p,r.] \/ {} \/ [.s,q.[ by Th28
      .= ].p,q.[ by A2,A3,A5,Th178;
  end;
end;
