Article L_HOSPIT, MML version 4.99.1005

:: L_HOSPIT:th 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
      st b1 is_continuous_in b2 &
         (for b3, b4 being Element of REAL
               st b3 < b2 & b2 < b4
            holds ex b5, b6 being Element of REAL st
               b3 < b5 & b5 < b2 & b5 in dom b1 & b6 < b4 & b2 < b6 & b6 in dom b1)
   holds b1 is_convergent_in b2;

:: L_HOSPIT:th 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
   b1 is_right_convergent_in b2 & lim_right(b1,b2) = b3
iff
   (for b4 being Element of REAL
          st b2 < b4
       holds ex b5 being Element of REAL st
          b5 < b4 & b2 < b5 & b5 in dom b1) &
    (for b4 being Function-like quasi_total Relation of NAT,REAL
          st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ right_open_halfline b2
       holds b1 * b4 is convergent & lim (b1 * b4) = b3);

:: L_HOSPIT:th 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2, b3 being Element of REAL holds
   b1 is_left_convergent_in b2 & lim_left(b1,b2) = b3
iff
   (for b4 being Element of REAL
          st b4 < b2
       holds ex b5 being Element of REAL st
          b4 < b5 & b5 < b2 & b5 in dom b1) &
    (for b4 being Function-like quasi_total Relation of NAT,REAL
          st b4 is convergent & lim b4 = b2 & rng b4 c= (dom b1) /\ halfline b2
       holds b1 * b4 is convergent & lim (b1 * b4) = b3);

:: L_HOSPIT:th 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
   st ex b3 being Neighbourhood of b2 st
        b3 \ {b2} c= dom b1
for b3, b4 being Element of REAL
      st b3 < b2 & b2 < b4
   holds ex b5, b6 being Element of REAL st
      b3 < b5 & b5 < b2 & b5 in dom b1 & b6 < b4 & b2 < b6 & b6 in dom b1;

:: L_HOSPIT:th 5
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Neighbourhood of b3 st
           b1 is_differentiable_on b4 &
            b2 is_differentiable_on b4 &
            b4 \ {b3} c= dom (b1 / b2) &
            b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            (b1 `| b4) / (b2 `| b4) is_divergent_to+infty_in b3
   holds b1 / b2 is_divergent_to+infty_in b3;

:: L_HOSPIT:th 6
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Neighbourhood of b3 st
           b1 is_differentiable_on b4 &
            b2 is_differentiable_on b4 &
            b4 \ {b3} c= dom (b1 / b2) &
            b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            (b1 `| b4) / (b2 `| b4) is_divergent_to-infty_in b3
   holds b1 / b2 is_divergent_to-infty_in b3;

:: L_HOSPIT:th 7
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Element of REAL st
           0 < b4 &
            b1 is_differentiable_on ].b3,b3 + b4.[ &
            b2 is_differentiable_on ].b3,b3 + b4.[ &
            ].b3,b3 + b4.[ c= dom (b1 / b2) &
            [.b3,b3 + b4.] c= dom ((b1 `| ].b3,b3 + b4.[) / (b2 `| ].b3,b3 + b4.[)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            b1 is_continuous_in b3 &
            b2 is_continuous_in b3 &
            (b1 `| ].b3,b3 + b4.[) / (b2 `| ].b3,b3 + b4.[) is_right_convergent_in b3
   holds b1 / b2 is_right_convergent_in b3 &
    (ex b4 being Element of REAL st
       0 < b4 &
        lim_right(b1 / b2,b3) = lim_right((b1 `| ].b3,b3 + b4.[) / (b2 `| ].b3,b3 + b4.[),b3));

:: L_HOSPIT:th 8
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Element of REAL st
           0 < b4 &
            b1 is_differentiable_on ].b3 - b4,b3.[ &
            b2 is_differentiable_on ].b3 - b4,b3.[ &
            ].b3 - b4,b3.[ c= dom (b1 / b2) &
            [.b3 - b4,b3.] c= dom ((b1 `| ].b3 - b4,b3.[) / (b2 `| ].b3 - b4,b3.[)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            b1 is_continuous_in b3 &
            b2 is_continuous_in b3 &
            (b1 `| ].b3 - b4,b3.[) / (b2 `| ].b3 - b4,b3.[) is_left_convergent_in b3
   holds b1 / b2 is_left_convergent_in b3 &
    (ex b4 being Element of REAL st
       0 < b4 &
        lim_left(b1 / b2,b3) = lim_left((b1 `| ].b3 - b4,b3.[) / (b2 `| ].b3 - b4,b3.[),b3));

:: L_HOSPIT:th 9
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Neighbourhood of b3 st
           b1 is_differentiable_on b4 &
            b2 is_differentiable_on b4 &
            b4 \ {b3} c= dom (b1 / b2) &
            b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            (b1 `| b4) / (b2 `| b4) is_convergent_in b3
   holds b1 / b2 is_convergent_in b3 &
    (ex b4 being Neighbourhood of b3 st
       lim(b1 / b2,b3) = lim((b1 `| b4) / (b2 `| b4),b3));

:: L_HOSPIT:th 10
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Element of REAL
      st ex b4 being Neighbourhood of b3 st
           b1 is_differentiable_on b4 &
            b2 is_differentiable_on b4 &
            b4 \ {b3} c= dom (b1 / b2) &
            b4 c= dom ((b1 `| b4) / (b2 `| b4)) &
            b1 . b3 = 0 &
            b2 . b3 = 0 &
            (b1 `| b4) / (b2 `| b4) is_continuous_in b3
   holds b1 / b2 is_convergent_in b3 &
    lim(b1 / b2,b3) = (diff(b1,b3)) / diff(b2,b3);