:: ROLLE semantic presentation
theorem Th1: :: ROLLE:1
theorem Th2: :: ROLLE:2
theorem Th3: :: ROLLE:3
theorem Th4: :: ROLLE:4
theorem Th5: :: ROLLE:5
for b
1, b
2 being
Real holds
( b
1 < b
2 implies for b
3, b
4 being
PartFunc of
REAL ,
REAL holds
not ( b
3 is_continuous_on [.b1,b2.] & b
3 is_differentiable_on ].b1,b2.[ & b
4 is_continuous_on [.b1,b2.] & b
4 is_differentiable_on ].b1,b2.[ & ( for b
5 being
Real holds
not ( b
5 in ].b1,b2.[ &
((b3 . b2) - (b3 . b1)) * (diff b4,b5) = ((b4 . b2) - (b4 . b1)) * (diff b3,b5) ) ) ) )
theorem Th6: :: ROLLE:6
for b
1, b
2 being
Real holds
( 0
< b
2 implies for b
3, b
4 being
PartFunc of
REAL ,
REAL holds
not ( b
3 is_continuous_on [.b1,(b1 + b2).] & b
3 is_differentiable_on ].b1,(b1 + b2).[ & b
4 is_continuous_on [.b1,(b1 + b2).] & b
4 is_differentiable_on ].b1,(b1 + b2).[ & ( for b
5 being
Real holds
not ( b
5 in ].b1,(b1 + b2).[ & not
diff b
4,b
5 <> 0 ) ) & ( for b
5 being
Real holds
not ( 0
< b
5 & b
5 < 1 &
((b3 . (b1 + b2)) - (b3 . b1)) / ((b4 . (b1 + b2)) - (b4 . b1)) = (diff b3,(b1 + (b5 * b2))) / (diff b4,(b1 + (b5 * b2))) ) ) ) )
theorem Th7: :: ROLLE:7
theorem Th8: :: ROLLE:8
for b
1, b
2 being
Real holds
( b
1 < b
2 implies for b
3, b
4 being
PartFunc of
REAL ,
REAL holds
( b
3 is_differentiable_on ].b1,b2.[ & b
4 is_differentiable_on ].b1,b2.[ & ( for b
5 being
Real holds
( b
5 in ].b1,b2.[ implies
diff b
3,b
5 = diff b
4,b
5 ) ) implies ( b
3 - b
4 is_constant_on ].b1,b2.[ & ex b
5 being
Real st
for b
6 being
Real holds
( b
6 in ].b1,b2.[ implies b
3 . b
6 = (b4 . b6) + b
5 ) ) ) )
theorem Th9: :: ROLLE:9
theorem Th10: :: ROLLE:10
theorem Th11: :: ROLLE:11
theorem Th12: :: ROLLE:12