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

theorem
  r <= s & s < t implies [.r,s.] /\ [.s,t.[ = {s}
proof
  assume that
A1: r <= s and
A2: s < t;
  now
    let x be object;
    hereby
      assume
A3:   x in [.r,s.] /\ [.s,t.[;
      then reconsider p = x as ExtReal;
A4:   p in [.r,s.] by A3,XBOOLE_0:def 4;
      p in [.s,t.[ by A3,XBOOLE_0:def 4;
      then
A5:   s <= p by Th3;
      p <= s by A4,Th1;
      hence x = s by A5,XXREAL_0:1;
    end;
    assume
A6: x = s;
A7: s in [.r,s.] by A1,Th1;
    s in [.s,t.[ by A2,Th3;
    hence x in [.r,s.] /\ [.s,t.[ by A6,A7,XBOOLE_0:def 4;
  end;
  hence thesis by TARSKI:def 1;
end;
