Article ROLLE, MML version 4.99.1005

:: ROLLE:th 1
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b2.] & b3 . b1 = b3 . b2 & b3 is_differentiable_on ].b1,b2.[
   holds ex b4 being Element of REAL st
      b4 in ].b1,b2.[ & diff(b3,b4) = 0;

:: ROLLE:th 2
theorem
for b1, b2 being Element of REAL
   st 0 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b1 + b2.] & b3 . b1 = b3 . (b1 + b2) & b3 is_differentiable_on ].b1,b1 + b2.[
   holds ex b4 being Element of REAL st
      0 < b4 & b4 < 1 & diff(b3,b1 + (b4 * b2)) = 0;

:: ROLLE:th 3
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b2.] & b3 is_differentiable_on ].b1,b2.[
   holds ex b4 being Element of REAL st
      b4 in ].b1,b2.[ &
       diff(b3,b4) = ((b3 . b2) - (b3 . b1)) / (b2 - b1);

:: ROLLE:th 4
theorem
for b1, b2 being Element of REAL
   st 0 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b1 + b2.] & b3 is_differentiable_on ].b1,b1 + b2.[
   holds ex b4 being Element of REAL st
      0 < b4 &
       b4 < 1 &
       b3 . (b1 + b2) = (b3 . b1) + (b2 * diff(b3,b1 + (b4 * b2)));

:: ROLLE:th 5
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b2.] & b3 is_differentiable_on ].b1,b2.[ & b4 is_continuous_on [.b1,b2.] & b4 is_differentiable_on ].b1,b2.[
   holds ex b5 being Element of REAL st
      b5 in ].b1,b2.[ &
       ((b3 . b2) - (b3 . b1)) * diff(b4,b5) = ((b4 . b2) - (b4 . b1)) * diff(b3,b5);

:: ROLLE:th 6
theorem
for b1, b2 being Element of REAL
   st 0 < b2
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_continuous_on [.b1,b1 + b2.] &
         b3 is_differentiable_on ].b1,b1 + b2.[ &
         b4 is_continuous_on [.b1,b1 + b2.] &
         b4 is_differentiable_on ].b1,b1 + b2.[ &
         (for b5 being Element of REAL
               st b5 in ].b1,b1 + b2.[
            holds diff(b4,b5) <> 0)
   holds ex b5 being Element of REAL st
      0 < b5 &
       b5 < 1 &
       ((b3 . (b1 + b2)) - (b3 . b1)) / ((b4 . (b1 + b2)) - (b4 . b1)) = (diff(b3,b1 + (b5 * b2))) / diff(b4,b1 + (b5 * b2));

:: ROLLE:th 7
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds diff(b3,b4) = 0)
   holds b3 is_constant_on ].b1,b2.[;

:: ROLLE:th 8
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3, b4 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         b4 is_differentiable_on ].b1,b2.[ &
         (for b5 being Element of REAL
               st b5 in ].b1,b2.[
            holds diff(b3,b5) = diff(b4,b5))
   holds b3 - b4 is_constant_on ].b1,b2.[ &
    (ex b5 being Element of REAL st
       for b6 being Element of REAL
             st b6 in ].b1,b2.[
          holds b3 . b6 = (b4 . b6) + b5);

:: ROLLE:th 9
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds 0 < diff(b3,b4))
   holds b3 is_increasing_on ].b1,b2.[;

:: ROLLE:th 10
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds diff(b3,b4) < 0)
   holds b3 is_decreasing_on ].b1,b2.[;

:: ROLLE:th 11
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds 0 <= diff(b3,b4))
   holds b3 is_non_decreasing_on ].b1,b2.[;

:: ROLLE:th 12
theorem
for b1, b2 being Element of REAL
   st b1 < b2
for b3 being Function-like Relation of REAL,REAL
      st b3 is_differentiable_on ].b1,b2.[ &
         (for b4 being Element of REAL
               st b4 in ].b1,b2.[
            holds diff(b3,b4) <= 0)
   holds b3 is_non_increasing_on ].b1,b2.[;