Article LIMFUNC4, MML version 4.99.1005
:: LIMFUNC4:th 1
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being set
st rng b3 c= (dom (b1 * b2)) /\ b4
holds rng b3 c= dom (b1 * b2) & rng b3 c= b4 & rng b3 c= dom b2 & rng b3 c= (dom b2) /\ b4 & rng (b2 * b3) c= dom b1;
:: LIMFUNC4:th 2
theorem
for b1, b2 being Function-like Relation of REAL,REAL
for b3 being Function-like quasi_total Relation of NAT,REAL
for b4 being set
st rng b3 c= (dom (b1 * b2)) \ b4
holds rng b3 c= dom (b1 * b2) & rng b3 c= dom b2 & rng b3 c= (dom b2) \ b4 & rng (b2 * b3) c= dom b1;
:: LIMFUNC4:th 3
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to+infty &
b2 is divergent_in+infty_to+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in+infty_to+infty;
:: LIMFUNC4:th 4
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to+infty &
b2 is divergent_in+infty_to-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in+infty_to-infty;
:: LIMFUNC4:th 5
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to-infty &
b2 is divergent_in-infty_to+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in+infty_to+infty;
:: LIMFUNC4:th 6
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to-infty &
b2 is divergent_in-infty_to-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in+infty_to-infty;
:: LIMFUNC4:th 7
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to+infty &
b2 is divergent_in+infty_to+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in-infty_to+infty;
:: LIMFUNC4:th 8
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to+infty &
b2 is divergent_in+infty_to-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in-infty_to-infty;
:: LIMFUNC4:th 9
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to-infty &
b2 is divergent_in-infty_to+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in-infty_to+infty;
:: LIMFUNC4:th 10
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to-infty &
b2 is divergent_in-infty_to-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is divergent_in-infty_to-infty;
:: LIMFUNC4:th 11
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to+infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC4:th 12
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to-infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC4:th 13
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to+infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC4:th 14
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to-infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC4:th 15
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to+infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC4:th 16
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to-infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC4:th 17
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to+infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC4:th 18
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to-infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC4:th 19
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_left_divergent_to+infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 < lim_left(b2,b1)))
holds b3 * b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC4:th 20
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_left_divergent_to-infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 < lim_left(b2,b1)))
holds b3 * b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC4:th 21
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_right_divergent_to+infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds lim_left(b2,b1) < b2 . b5))
holds b3 * b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC4:th 22
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_right_divergent_to-infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds lim_left(b2,b1) < b2 . b5))
holds b3 * b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC4:th 23
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_right_divergent_to+infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds lim_right(b2,b1) < b2 . b5))
holds b3 * b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC4:th 24
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_right_divergent_to-infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds lim_right(b2,b1) < b2 . b5))
holds b3 * b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC4:th 25
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_left_divergent_to+infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 < lim_right(b2,b1)))
holds b3 * b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC4:th 26
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_left_divergent_to-infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 < lim_right(b2,b1)))
holds b3 * b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC4:th 27
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_left_divergent_to+infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 < lim_in+infty b1)
holds b2 * b1 is divergent_in+infty_to+infty;
:: LIMFUNC4:th 28
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_left_divergent_to-infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 < lim_in+infty b1)
holds b2 * b1 is divergent_in+infty_to-infty;
:: LIMFUNC4:th 29
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_right_divergent_to+infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds lim_in+infty b1 < b1 . b4)
holds b2 * b1 is divergent_in+infty_to+infty;
:: LIMFUNC4:th 30
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_right_divergent_to-infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds lim_in+infty b1 < b1 . b4)
holds b2 * b1 is divergent_in+infty_to-infty;
:: LIMFUNC4:th 31
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_left_divergent_to+infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 < lim_in-infty b1)
holds b2 * b1 is divergent_in-infty_to+infty;
:: LIMFUNC4:th 32
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_left_divergent_to-infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 < lim_in-infty b1)
holds b2 * b1 is divergent_in-infty_to-infty;
:: LIMFUNC4:th 33
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_right_divergent_to+infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds lim_in-infty b1 < b1 . b4)
holds b2 * b1 is divergent_in-infty_to+infty;
:: LIMFUNC4:th 34
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_right_divergent_to-infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds lim_in-infty b1 < b1 . b4)
holds b2 * b1 is divergent_in-infty_to-infty;
:: LIMFUNC4:th 35
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to+infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_divergent_to+infty_in b1;
:: LIMFUNC4:th 36
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to+infty_in b1 &
b3 is divergent_in+infty_to-infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_divergent_to-infty_in b1;
:: LIMFUNC4:th 37
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to+infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_divergent_to+infty_in b1;
:: LIMFUNC4:th 38
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to-infty_in b1 &
b3 is divergent_in-infty_to-infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_divergent_to-infty_in b1;
:: LIMFUNC4:th 39
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_divergent_to+infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 <> lim(b2,b1)))
holds b3 * b2 is_divergent_to+infty_in b1;
:: LIMFUNC4:th 40
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_divergent_to-infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 <> lim(b2,b1)))
holds b3 * b2 is_divergent_to-infty_in b1;
:: LIMFUNC4:th 41
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_right_divergent_to+infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds lim(b2,b1) < b2 . b5))
holds b3 * b2 is_divergent_to+infty_in b1;
:: LIMFUNC4:th 42
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_right_divergent_to-infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds lim(b2,b1) < b2 . b5))
holds b3 * b2 is_divergent_to-infty_in b1;
:: LIMFUNC4:th 43
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_divergent_to+infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 <> lim_right(b2,b1)))
holds b3 * b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC4:th 44
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_divergent_to-infty_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 <> lim_right(b2,b1)))
holds b3 * b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC4:th 45
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_divergent_to+infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 <> lim_in+infty b1)
holds b2 * b1 is divergent_in+infty_to+infty;
:: LIMFUNC4:th 46
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_divergent_to-infty_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 <> lim_in+infty b1)
holds b2 * b1 is divergent_in+infty_to-infty;
:: LIMFUNC4:th 47
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_divergent_to+infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 <> lim_in-infty b1)
holds b2 * b1 is divergent_in-infty_to+infty;
:: LIMFUNC4:th 48
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_divergent_to-infty_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 <> lim_in-infty b1)
holds b2 * b1 is divergent_in-infty_to-infty;
:: LIMFUNC4:th 49
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_left_divergent_to+infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 < lim(b2,b1)))
holds b3 * b2 is_divergent_to+infty_in b1;
:: LIMFUNC4:th 50
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_left_divergent_to-infty_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 < lim(b2,b1)))
holds b3 * b2 is_divergent_to-infty_in b1;
:: LIMFUNC4:th 51
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_divergent_to+infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 <> lim_left(b2,b1)))
holds b3 * b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC4:th 52
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_divergent_to-infty_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 <> lim_left(b2,b1)))
holds b3 * b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC4:th 53
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to+infty &
b2 is convergent_in+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is convergent_in+infty & lim_in+infty (b2 * b1) = lim_in+infty b2;
:: LIMFUNC4:th 54
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in+infty_to-infty &
b2 is convergent_in-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1))
holds b2 * b1 is convergent_in+infty & lim_in+infty (b2 * b1) = lim_in-infty b2;
:: LIMFUNC4:th 55
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to+infty &
b2 is convergent_in+infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is convergent_in-infty & lim_in-infty (b2 * b1) = lim_in+infty b2;
:: LIMFUNC4:th 56
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is divergent_in-infty_to-infty &
b2 is convergent_in-infty &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1))
holds b2 * b1 is convergent_in-infty & lim_in-infty (b2 * b1) = lim_in-infty b2;
:: LIMFUNC4:th 57
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to+infty_in b1 &
b3 is convergent_in+infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_convergent_in b1 & lim_left(b3 * b2,b1) = lim_in+infty b3;
:: LIMFUNC4:th 58
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_divergent_to-infty_in b1 &
b3 is convergent_in-infty &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2))
holds b3 * b2 is_left_convergent_in b1 & lim_left(b3 * b2,b1) = lim_in-infty b3;
:: LIMFUNC4:th 59
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to+infty_in b1 &
b3 is convergent_in+infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_convergent_in b1 & lim_right(b3 * b2,b1) = lim_in+infty b3;
:: LIMFUNC4:th 60
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_divergent_to-infty_in b1 &
b3 is convergent_in-infty &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2))
holds b3 * b2 is_right_convergent_in b1 & lim_right(b3 * b2,b1) = lim_in-infty b3;
:: LIMFUNC4:th 61
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_left_convergent_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 < lim_left(b2,b1)))
holds b3 * b2 is_left_convergent_in b1 &
lim_left(b3 * b2,b1) = lim_left(b3,lim_left(b2,b1));
:: LIMFUNC4:th 62
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_right_convergent_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds lim_right(b2,b1) < b2 . b5))
holds b3 * b2 is_right_convergent_in b1 &
lim_right(b3 * b2,b1) = lim_right(b3,lim_right(b2,b1));
:: LIMFUNC4:th 63
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_right_convergent_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds lim_left(b2,b1) < b2 . b5))
holds b3 * b2 is_left_convergent_in b1 &
lim_left(b3 * b2,b1) = lim_right(b3,lim_left(b2,b1));
:: LIMFUNC4:th 64
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_left_convergent_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 < lim_right(b2,b1)))
holds b3 * b2 is_right_convergent_in b1 &
lim_right(b3 * b2,b1) = lim_left(b3,lim_right(b2,b1));
:: LIMFUNC4:th 65
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_left_convergent_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 < lim_in+infty b1)
holds b2 * b1 is convergent_in+infty &
lim_in+infty (b2 * b1) = lim_left(b2,lim_in+infty b1);
:: LIMFUNC4:th 66
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_right_convergent_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds lim_in+infty b1 < b1 . b4)
holds b2 * b1 is convergent_in+infty &
lim_in+infty (b2 * b1) = lim_right(b2,lim_in+infty b1);
:: LIMFUNC4:th 67
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_left_convergent_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 < lim_in-infty b1)
holds b2 * b1 is convergent_in-infty &
lim_in-infty (b2 * b1) = lim_left(b2,lim_in-infty b1);
:: LIMFUNC4:th 68
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_right_convergent_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds lim_in-infty b1 < b1 . b4)
holds b2 * b1 is convergent_in-infty &
lim_in-infty (b2 * b1) = lim_right(b2,lim_in-infty b1);
:: LIMFUNC4:th 69
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to+infty_in b1 &
b3 is convergent_in+infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_convergent_in b1 & lim(b3 * b2,b1) = lim_in+infty b3;
:: LIMFUNC4:th 70
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_divergent_to-infty_in b1 &
b3 is convergent_in-infty &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2))
holds b3 * b2 is_convergent_in b1 & lim(b3 * b2,b1) = lim_in-infty b3;
:: LIMFUNC4:th 71
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in+infty &
b2 is_convergent_in lim_in+infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b3 < b4 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ right_open_halfline b3
holds b1 . b4 <> lim_in+infty b1)
holds b2 * b1 is convergent_in+infty &
lim_in+infty (b2 * b1) = lim(b2,lim_in+infty b1);
:: LIMFUNC4:th 72
theorem
for b1, b2 being Function-like Relation of REAL,REAL
st b1 is convergent_in-infty &
b2 is_convergent_in lim_in-infty b1 &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b3 & b4 in dom (b2 * b1)) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st b4 in (dom b1) /\ halfline b3
holds b1 . b4 <> lim_in-infty b1)
holds b2 * b1 is convergent_in-infty &
lim_in-infty (b2 * b1) = lim(b2,lim_in-infty b1);
:: LIMFUNC4:th 73
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_left_convergent_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 < lim(b2,b1)))
holds b3 * b2 is_convergent_in b1 &
lim(b3 * b2,b1) = lim_left(b3,lim(b2,b1));
:: LIMFUNC4:th 74
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_convergent_in lim_left(b2,b1) &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 <> lim_left(b2,b1)))
holds b3 * b2 is_left_convergent_in b1 &
lim_left(b3 * b2,b1) = lim(b3,lim_left(b2,b1));
:: LIMFUNC4:th 75
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_right_convergent_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds lim(b2,b1) < b2 . b5))
holds b3 * b2 is_convergent_in b1 &
lim(b3 * b2,b1) = lim_right(b3,lim(b2,b1));
:: LIMFUNC4:th 76
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_convergent_in lim_right(b2,b1) &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 <> lim_right(b2,b1)))
holds b3 * b2 is_right_convergent_in b1 &
lim_right(b3 * b2,b1) = lim(b3,lim_right(b2,b1));
:: LIMFUNC4:th 77
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_convergent_in b1 &
b3 is_convergent_in lim(b2,b1) &
(for b4, b5 being Element of REAL
st b4 < b1 & b1 < b5
holds ex b6, b7 being Element of REAL st
b4 < b6 & b6 < b1 & b6 in dom (b3 * b2) & b7 < b5 & b1 < b7 & b7 in dom (b3 * b2)) &
(ex b4 being Element of REAL st
0 < b4 &
(for b5 being Element of REAL
st b5 in (dom b2) /\ (].b1 - b4,b1.[ \/ ].b1,b1 + b4.[)
holds b2 . b5 <> lim(b2,b1)))
holds b3 * b2 is_convergent_in b1 &
lim(b3 * b2,b1) = lim(b3,lim(b2,b1));