Article LIMFUNC2, MML version 4.99.1005
:: LIMFUNC2:th 1
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & b1 < lim b2
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b1 < b2 . b4;
:: LIMFUNC2:th 2
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & lim b2 < b1
holds ex b3 being Element of NAT st
for b4 being Element of NAT
st b3 <= b4
holds b2 . b4 < b1;
:: LIMFUNC2:th 3
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st 0 < b1 & ].b2 - b1,b2.[ c= dom b3
for b4 being Element of REAL
st b4 < b2
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b2 & b5 in dom b3;
:: LIMFUNC2:th 4
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st 0 < b1 & ].b2,b2 + b1.[ c= dom b3
for b4 being Element of REAL
st b2 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b2 < b5 & b5 in dom b3;
:: LIMFUNC2:th 5
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
st for b4 being Element of NAT holds
b1 - (1 / (b4 + 1)) < b2 . b4 &
b2 . b4 < b1 &
b2 . b4 in dom b3
holds b2 is convergent & lim b2 = b1 & rng b2 c= dom b3 & rng b2 c= (dom b3) /\ halfline b1;
:: LIMFUNC2:th 6
theorem
for b1 being Element of REAL
for b2 being Function-like quasi_total Relation of NAT,REAL
for b3 being Function-like Relation of REAL,REAL
st for b4 being Element of NAT holds
b1 < b2 . b4 &
b2 . b4 < b1 + (1 / (b4 + 1)) &
b2 . b4 in dom b3
holds b2 is convergent & lim b2 = b1 & rng b2 c= dom b3 & rng b2 c= (dom b3) /\ right_open_halfline b1;
:: LIMFUNC2:prednot 1 => LIMFUNC2:pred 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_left_convergent_in A2 means
(for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(ex b1 being Element of REAL st
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ halfline a2
holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;
:: LIMFUNC2:dfs 1
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_left_convergent_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(ex b1 being Element of REAL st
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ halfline a2
holds a1 * b2 is convergent & lim (a1 * b2) = b1);
:: LIMFUNC2:def 1
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_left_convergent_in b2
iff
(for b3 being Element of REAL
st b3 < b2
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b2 & b4 in dom b1) &
(ex b3 being Element of REAL st
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);
:: LIMFUNC2:prednot 2 => LIMFUNC2:pred 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_left_divergent_to+infty_in A2 means
(for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
holds a1 * b1 is divergent_to+infty);
end;
:: LIMFUNC2:dfs 2
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_left_divergent_to+infty_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
holds a1 * b1 is divergent_to+infty);
:: LIMFUNC2:def 2
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_left_divergent_to+infty_in b2
iff
(for b3 being Element of REAL
st b3 < b2
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b2 & b4 in dom b1) &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ halfline b2
holds b1 * b3 is divergent_to+infty);
:: LIMFUNC2:prednot 3 => LIMFUNC2:pred 3
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_left_divergent_to-infty_in A2 means
(for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
holds a1 * b1 is divergent_to-infty);
end;
:: LIMFUNC2:dfs 3
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_left_divergent_to-infty_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st b1 < a2
holds ex b2 being Element of REAL st
b1 < b2 & b2 < a2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
holds a1 * b1 is divergent_to-infty);
:: LIMFUNC2:def 3
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_left_divergent_to-infty_in b2
iff
(for b3 being Element of REAL
st b3 < b2
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b2 & b4 in dom b1) &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ halfline b2
holds b1 * b3 is divergent_to-infty);
:: LIMFUNC2:prednot 4 => LIMFUNC2:pred 4
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_right_convergent_in A2 means
(for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(ex b1 being Element of REAL st
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ right_open_halfline a2
holds a1 * b2 is convergent & lim (a1 * b2) = b1);
end;
:: LIMFUNC2:dfs 4
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_right_convergent_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(ex b1 being Element of REAL st
for b2 being Function-like quasi_total Relation of NAT,REAL
st b2 is convergent & lim b2 = a2 & rng b2 c= (dom a1) /\ right_open_halfline a2
holds a1 * b2 is convergent & lim (a1 * b2) = b1);
:: LIMFUNC2:def 4
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_right_convergent_in b2
iff
(for b3 being Element of REAL
st b2 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b2 < b4 & b4 in dom b1) &
(ex b3 being Element of REAL st
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);
:: LIMFUNC2:prednot 5 => LIMFUNC2:pred 5
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_right_divergent_to+infty_in A2 means
(for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
holds a1 * b1 is divergent_to+infty);
end;
:: LIMFUNC2:dfs 5
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_right_divergent_to+infty_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
holds a1 * b1 is divergent_to+infty);
:: LIMFUNC2:def 5
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_right_divergent_to+infty_in b2
iff
(for b3 being Element of REAL
st b2 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b2 < b4 & b4 in dom b1) &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ right_open_halfline b2
holds b1 * b3 is divergent_to+infty);
:: LIMFUNC2:prednot 6 => LIMFUNC2:pred 6
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
pred A1 is_right_divergent_to-infty_in A2 means
(for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
holds a1 * b1 is divergent_to-infty);
end;
:: LIMFUNC2:dfs 6
definiens
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
To prove
a1 is_right_divergent_to-infty_in a2
it is sufficient to prove
thus (for b1 being Element of REAL
st a2 < b1
holds ex b2 being Element of REAL st
b2 < b1 & a2 < b2 & b2 in dom a1) &
(for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
holds a1 * b1 is divergent_to-infty);
:: LIMFUNC2:def 6
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL holds
b1 is_right_divergent_to-infty_in b2
iff
(for b3 being Element of REAL
st b2 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b2 < b4 & b4 in dom b1) &
(for b3 being Function-like quasi_total Relation of NAT,REAL
st b3 is convergent & lim b3 = b2 & rng b3 c= (dom b1) /\ right_open_halfline b2
holds b1 * b3 is divergent_to-infty);
:: LIMFUNC2:th 13
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_left_convergent_in b1
iff
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
b5 < b1 &
(for b6 being Element of REAL
st b5 < b6 & b6 < b1 & b6 in dom b2
holds abs ((b2 . b6) - b3) < b4));
:: LIMFUNC2:th 14
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_left_divergent_to+infty_in b1
iff
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2) &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b1 &
(for b5 being Element of REAL
st b4 < b5 & b5 < b1 & b5 in dom b2
holds b3 < b2 . b5));
:: LIMFUNC2:th 15
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_left_divergent_to-infty_in b1
iff
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2) &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b4 < b1 &
(for b5 being Element of REAL
st b4 < b5 & b5 < b1 & b5 in dom b2
holds b2 . b5 < b3));
:: LIMFUNC2:th 16
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_right_convergent_in b1
iff
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2) &
(ex b3 being Element of REAL st
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
b1 < b5 &
(for b6 being Element of REAL
st b6 < b5 & b1 < b6 & b6 in dom b2
holds abs ((b2 . b6) - b3) < b4));
:: LIMFUNC2:th 17
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_right_divergent_to+infty_in b1
iff
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2) &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b1 < b4 &
(for b5 being Element of REAL
st b5 < b4 & b1 < b5 & b5 in dom b2
holds b3 < b2 . b5));
:: LIMFUNC2:th 18
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL holds
b2 is_right_divergent_to-infty_in b1
iff
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2) &
(for b3 being Element of REAL holds
ex b4 being Element of REAL st
b1 < b4 &
(for b5 being Element of REAL
st b5 < b4 & b1 < b5 & b5 in dom b2
holds b2 . b5 < b3));
:: LIMFUNC2:th 19
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_left_divergent_to+infty_in 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 b2) /\ dom b3)
holds b2 + b3 is_left_divergent_to+infty_in b1 & b2 (#) b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 20
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_left_divergent_to-infty_in 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 b2) /\ dom b3)
holds b2 + b3 is_left_divergent_to-infty_in b1 & b2 (#) b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 21
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_right_divergent_to+infty_in 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 b2) /\ dom b3)
holds b2 + b3 is_right_divergent_to+infty_in b1 & b2 (#) b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 22
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_right_divergent_to-infty_in 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 b2) /\ dom b3)
holds b2 + b3 is_right_divergent_to-infty_in b1 & b2 (#) b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 23
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 &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b2 + b3)) &
(ex b4 being Element of REAL st
0 < b4 & b3 is_bounded_below_on ].b1 - b4,b1.[)
holds b2 + b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 24
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 &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b2 (#) b3)) &
(ex b4, b5 being Element of REAL st
0 < b4 &
0 < b5 &
(for b6 being Element of REAL
st b6 in (dom b3) /\ ].b1 - b4,b1.[
holds b5 <= b3 . b6))
holds b2 (#) b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 25
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 &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b2 + b3)) &
(ex b4 being Element of REAL st
0 < b4 & b3 is_bounded_below_on ].b1,b1 + b4.[)
holds b2 + b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 26
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 &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b2 (#) b3)) &
(ex b4, b5 being Element of REAL st
0 < b4 &
0 < b5 &
(for b6 being Element of REAL
st b6 in (dom b3) /\ ].b1,b1 + b4.[
holds b5 <= b3 . b6))
holds b2 (#) b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 27
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
(b3 is_left_divergent_to+infty_in b1 & 0 < b2 implies b2 (#) b3 is_left_divergent_to+infty_in b1) &
(b3 is_left_divergent_to+infty_in b1 & b2 < 0 implies b2 (#) b3 is_left_divergent_to-infty_in b1) &
(b3 is_left_divergent_to-infty_in b1 & 0 < b2 implies b2 (#) b3 is_left_divergent_to-infty_in b1) &
(b3 is_left_divergent_to-infty_in b1 & b2 < 0 implies b2 (#) b3 is_left_divergent_to+infty_in b1);
:: LIMFUNC2:th 28
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL holds
(b3 is_right_divergent_to+infty_in b1 & 0 < b2 implies b2 (#) b3 is_right_divergent_to+infty_in b1) &
(b3 is_right_divergent_to+infty_in b1 & b2 < 0 implies b2 (#) b3 is_right_divergent_to-infty_in b1) &
(b3 is_right_divergent_to-infty_in b1 & 0 < b2 implies b2 (#) b3 is_right_divergent_to-infty_in b1) &
(b3 is_right_divergent_to-infty_in b1 & b2 < 0 implies b2 (#) b3 is_right_divergent_to+infty_in b1);
:: LIMFUNC2:th 29
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (b2 is_left_divergent_to+infty_in b1 or b2 is_left_divergent_to-infty_in b1)
holds abs b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 30
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (b2 is_right_divergent_to+infty_in b1 or b2 is_right_divergent_to-infty_in b1)
holds abs b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 31
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_non_decreasing_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[) &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2)
holds b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 32
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_increasing_on ].b1 - b3,b1.[ & not b2 is_bounded_above_on ].b1 - b3,b1.[) &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2)
holds b2 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 33
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_non_increasing_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[) &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2)
holds b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 34
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_decreasing_on ].b1 - b3,b1.[ & not b2 is_bounded_below_on ].b1 - b3,b1.[) &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2)
holds b2 is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 35
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_non_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2)
holds b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 36
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_above_on ].b1,b1 + b3.[) &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2)
holds b2 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 37
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_non_decreasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2)
holds b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC2:th 38
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (ex b3 being Element of REAL st
0 < b3 & b2 is_increasing_on ].b1,b1 + b3.[ & not b2 is_bounded_below_on ].b1,b1 + b3.[) &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2)
holds b2 is_right_divergent_to-infty_in b1;
:: LIMFUNC2:th 39
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 &
(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) &
(ex b4 being Element of REAL st
0 < b4 &
(dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1 - b4,b1.[
holds b2 . b5 <= b3 . b5))
holds b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 40
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 &
(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) &
(ex b4 being Element of REAL st
0 < b4 &
(dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1 - b4,b1.[
holds b3 . b5 <= b2 . b5))
holds b3 is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 41
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 &
(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) &
(ex b4 being Element of REAL st
0 < b4 &
(dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1,b1 + b4.[
holds b2 . b5 <= b3 . b5))
holds b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 42
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 &
(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) &
(ex b4 being Element of REAL st
0 < b4 &
(dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1,b1 + b4.[
holds b3 . b5 <= b2 . b5))
holds b3 is_right_divergent_to-infty_in b1;
:: LIMFUNC2:th 43
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 &
(ex b4 being Element of REAL st
0 < b4 &
].b1 - b4,b1.[ c= (dom b3) /\ dom b2 &
(for b5 being Element of REAL
st b5 in ].b1 - b4,b1.[
holds b2 . b5 <= b3 . b5))
holds b3 is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 44
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 &
(ex b4 being Element of REAL st
0 < b4 &
].b1 - b4,b1.[ c= (dom b3) /\ dom b2 &
(for b5 being Element of REAL
st b5 in ].b1 - b4,b1.[
holds b3 . b5 <= b2 . b5))
holds b3 is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 45
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 &
(ex b4 being Element of REAL st
0 < b4 &
].b1,b1 + b4.[ c= (dom b3) /\ dom b2 &
(for b5 being Element of REAL
st b5 in ].b1,b1 + b4.[
holds b2 . b5 <= b3 . b5))
holds b3 is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 46
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 &
(ex b4 being Element of REAL st
0 < b4 &
].b1,b1 + b4.[ c= (dom b3) /\ dom b2 &
(for b5 being Element of REAL
st b5 in ].b1,b1 + b4.[
holds b3 . b5 <= b2 . b5))
holds b3 is_right_divergent_to-infty_in b1;
:: LIMFUNC2:funcnot 1 => LIMFUNC2:func 1
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
assume a1 is_left_convergent_in a2;
func lim_left(A1,A2) -> Element of REAL means
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ halfline a2
holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;
:: LIMFUNC2:def 7
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
st b1 is_left_convergent_in b2
for b3 being Element of REAL holds
b3 = lim_left(b1,b2)
iff
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;
:: LIMFUNC2:funcnot 2 => LIMFUNC2:func 2
definition
let a1 be Function-like Relation of REAL,REAL;
let a2 be Element of REAL;
assume a1 is_right_convergent_in a2;
func lim_right(A1,A2) -> Element of REAL means
for b1 being Function-like quasi_total Relation of NAT,REAL
st b1 is convergent & lim b1 = a2 & rng b1 c= (dom a1) /\ right_open_halfline a2
holds a1 * b1 is convergent & lim (a1 * b1) = it;
end;
:: LIMFUNC2:def 8
theorem
for b1 being Function-like Relation of REAL,REAL
for b2 being Element of REAL
st b1 is_right_convergent_in b2
for b3 being Element of REAL holds
b3 = lim_right(b1,b2)
iff
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;
:: LIMFUNC2:th 49
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_left_convergent_in b1
holds lim_left(b3,b1) = b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
b5 < b1 &
(for b6 being Element of REAL
st b5 < b6 & b6 < b1 & b6 in dom b3
holds abs ((b3 . b6) - b2) < b4);
:: LIMFUNC2:th 50
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_right_convergent_in b1
holds lim_right(b3,b1) = b2
iff
for b4 being Element of REAL
st 0 < b4
holds ex b5 being Element of REAL st
b1 < b5 &
(for b6 being Element of REAL
st b6 < b5 & b1 < b6 & b6 in dom b3
holds abs ((b3 . b6) - b2) < b4);
:: LIMFUNC2:th 51
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_left_convergent_in b1
holds b2 (#) b3 is_left_convergent_in b1 &
lim_left(b2 (#) b3,b1) = b2 * lim_left(b3,b1);
:: LIMFUNC2:th 52
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1
holds - b2 is_left_convergent_in b1 & lim_left(- b2,b1) = - lim_left(b2,b1);
:: LIMFUNC2:th 53
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 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 (b2 + b3))
holds b2 + b3 is_left_convergent_in b1 &
lim_left(b2 + b3,b1) = (lim_left(b2,b1)) + lim_left(b3,b1);
:: LIMFUNC2:th 54
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 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 (b2 - b3))
holds b2 - b3 is_left_convergent_in b1 &
lim_left(b2 - b3,b1) = (lim_left(b2,b1)) - lim_left(b3,b1);
:: LIMFUNC2:th 55
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 & b2 " {0} = {} & lim_left(b2,b1) <> 0
holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = (lim_left(b2,b1)) ";
:: LIMFUNC2:th 56
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1
holds abs b2 is_left_convergent_in b1 &
lim_left(abs b2,b1) = abs lim_left(b2,b1);
:: LIMFUNC2:th 57
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) <> 0 &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0)
holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = (lim_left(b2,b1)) ";
:: LIMFUNC2:th 58
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 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 (b2 (#) b3))
holds b2 (#) b3 is_left_convergent_in b1 &
lim_left(b2 (#) b3,b1) = (lim_left(b2,b1)) * lim_left(b3,b1);
:: LIMFUNC2:th 59
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 b1 &
lim_left(b3,b1) <> 0 &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b2 / b3))
holds b2 / b3 is_left_convergent_in b1 &
lim_left(b2 / b3,b1) = (lim_left(b2,b1)) / lim_left(b3,b1);
:: LIMFUNC2:th 60
theorem
for b1, b2 being Element of REAL
for b3 being Function-like Relation of REAL,REAL
st b3 is_right_convergent_in b1
holds b2 (#) b3 is_right_convergent_in b1 &
lim_right(b2 (#) b3,b1) = b2 * lim_right(b3,b1);
:: LIMFUNC2:th 61
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1
holds - b2 is_right_convergent_in b1 & lim_right(- b2,b1) = - lim_right(b2,b1);
:: LIMFUNC2: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 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 (b2 + b3))
holds b2 + b3 is_right_convergent_in b1 &
lim_right(b2 + b3,b1) = (lim_right(b2,b1)) + lim_right(b3,b1);
:: LIMFUNC2:th 63
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 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 (b2 - b3))
holds b2 - b3 is_right_convergent_in b1 &
lim_right(b2 - b3,b1) = (lim_right(b2,b1)) - lim_right(b3,b1);
:: LIMFUNC2:th 64
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 & b2 " {0} = {} & lim_right(b2,b1) <> 0
holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = (lim_right(b2,b1)) ";
:: LIMFUNC2:th 65
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1
holds abs b2 is_right_convergent_in b1 &
lim_right(abs b2,b1) = abs lim_right(b2,b1);
:: LIMFUNC2:th 66
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) <> 0 &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0)
holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = (lim_right(b2,b1)) ";
:: LIMFUNC2:th 67
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 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 (b2 (#) b3))
holds b2 (#) b3 is_right_convergent_in b1 &
lim_right(b2 (#) b3,b1) = (lim_right(b2,b1)) * lim_right(b3,b1);
:: LIMFUNC2:th 68
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 b1 &
lim_right(b3,b1) <> 0 &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b2 / b3))
holds b2 / b3 is_right_convergent_in b1 &
lim_right(b2 / b3,b1) = (lim_right(b2,b1)) / lim_right(b3,b1);
:: LIMFUNC2:th 69
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) = 0 &
(for b4 being Element of REAL
st b4 < b1
holds ex b5 being Element of REAL st
b4 < b5 & b5 < b1 & b5 in dom (b2 (#) b3)) &
(ex b4 being Element of REAL st
0 < b4 & b3 is_bounded_on ].b1 - b4,b1.[)
holds b2 (#) b3 is_left_convergent_in b1 & lim_left(b2 (#) b3,b1) = 0;
:: LIMFUNC2:th 70
theorem
for b1 being Element of REAL
for b2, b3 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) = 0 &
(for b4 being Element of REAL
st b1 < b4
holds ex b5 being Element of REAL st
b5 < b4 & b1 < b5 & b5 in dom (b2 (#) b3)) &
(ex b4 being Element of REAL st
0 < b4 & b3 is_bounded_on ].b1,b1 + b4.[)
holds b2 (#) b3 is_right_convergent_in b1 & lim_right(b2 (#) b3,b1) = 0;
:: LIMFUNC2:th 71
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_left_convergent_in b1 &
lim_left(b2,b1) = lim_left(b3,b1) &
(for b5 being Element of REAL
st b5 < b1
holds ex b6 being Element of REAL st
b5 < b6 & b6 < b1 & b6 in dom b4) &
(ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of REAL
st b6 in (dom b4) /\ ].b1 - b5,b1.[
holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6) &
((dom b2) /\ ].b1 - b5,b1.[ c= (dom b3) /\ ].b1 - b5,b1.[ &
(dom b4) /\ ].b1 - b5,b1.[ c= (dom b2) /\ ].b1 - b5,b1.[ or (dom b3) /\ ].b1 - b5,b1.[ c= (dom b2) /\ ].b1 - b5,b1.[ &
(dom b4) /\ ].b1 - b5,b1.[ c= (dom b3) /\ ].b1 - b5,b1.[))
holds b4 is_left_convergent_in b1 & lim_left(b4,b1) = lim_left(b2,b1);
:: LIMFUNC2:th 72
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
b3 is_left_convergent_in b1 &
lim_left(b2,b1) = lim_left(b3,b1) &
(ex b5 being Element of REAL st
0 < b5 &
].b1 - b5,b1.[ c= ((dom b2) /\ dom b3) /\ dom b4 &
(for b6 being Element of REAL
st b6 in ].b1 - b5,b1.[
holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6))
holds b4 is_left_convergent_in b1 & lim_left(b4,b1) = lim_left(b2,b1);
:: LIMFUNC2:th 73
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_right_convergent_in b1 &
lim_right(b2,b1) = lim_right(b3,b1) &
(for b5 being Element of REAL
st b1 < b5
holds ex b6 being Element of REAL st
b6 < b5 & b1 < b6 & b6 in dom b4) &
(ex b5 being Element of REAL st
0 < b5 &
(for b6 being Element of REAL
st b6 in (dom b4) /\ ].b1,b1 + b5.[
holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6) &
((dom b2) /\ ].b1,b1 + b5.[ c= (dom b3) /\ ].b1,b1 + b5.[ &
(dom b4) /\ ].b1,b1 + b5.[ c= (dom b2) /\ ].b1,b1 + b5.[ or (dom b3) /\ ].b1,b1 + b5.[ c= (dom b2) /\ ].b1,b1 + b5.[ &
(dom b4) /\ ].b1,b1 + b5.[ c= (dom b3) /\ ].b1,b1 + b5.[))
holds b4 is_right_convergent_in b1 & lim_right(b4,b1) = lim_right(b2,b1);
:: LIMFUNC2:th 74
theorem
for b1 being Element of REAL
for b2, b3, b4 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
b3 is_right_convergent_in b1 &
lim_right(b2,b1) = lim_right(b3,b1) &
(ex b5 being Element of REAL st
0 < b5 &
].b1,b1 + b5.[ c= ((dom b2) /\ dom b3) /\ dom b4 &
(for b6 being Element of REAL
st b6 in ].b1,b1 + b5.[
holds b2 . b6 <= b4 . b6 & b4 . b6 <= b3 . b6))
holds b4 is_right_convergent_in b1 & lim_right(b4,b1) = lim_right(b2,b1);
:: LIMFUNC2:th 75
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 b1 &
(ex b4 being Element of REAL st
0 < b4 &
((dom b2) /\ ].b1 - b4,b1.[ c= (dom b3) /\ ].b1 - b4,b1.[ &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1 - b4,b1.[
holds b2 . b5 <= b3 . b5) or (dom b3) /\ ].b1 - b4,b1.[ c= (dom b2) /\ ].b1 - b4,b1.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1 - b4,b1.[
holds b2 . b5 <= b3 . b5)))
holds lim_left(b2,b1) <= lim_left(b3,b1);
:: LIMFUNC2: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_right_convergent_in b1 &
(ex b4 being Element of REAL st
0 < b4 &
((dom b2) /\ ].b1,b1 + b4.[ c= (dom b3) /\ ].b1,b1 + b4.[ &
(for b5 being Element of REAL
st b5 in (dom b2) /\ ].b1,b1 + b4.[
holds b2 . b5 <= b3 . b5) or (dom b3) /\ ].b1,b1 + b4.[ c= (dom b2) /\ ].b1,b1 + b4.[ &
(for b5 being Element of REAL
st b5 in (dom b3) /\ ].b1,b1 + b4.[
holds b2 . b5 <= b3 . b5)))
holds lim_right(b2,b1) <= lim_right(b3,b1);
:: LIMFUNC2:th 77
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (b2 is_left_divergent_to+infty_in b1 or b2 is_left_divergent_to-infty_in b1) &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0)
holds b2 ^ is_left_convergent_in b1 & lim_left(b2 ^,b1) = 0;
:: LIMFUNC2:th 78
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st (b2 is_right_divergent_to+infty_in b1 or b2 is_right_divergent_to-infty_in b1) &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0)
holds b2 ^ is_right_convergent_in b1 & lim_right(b2 ^,b1) = 0;
:: LIMFUNC2:th 79
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) = 0 &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1 - b3,b1.[
holds 0 < b2 . b4))
holds b2 ^ is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 80
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) = 0 &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1 - b3,b1.[
holds b2 . b4 < 0))
holds b2 ^ is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 81
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) = 0 &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1,b1 + b3.[
holds 0 < b2 . b4))
holds b2 ^ is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 82
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) = 0 &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1,b1 + b3.[
holds b2 . b4 < 0))
holds b2 ^ is_right_divergent_to-infty_in b1;
:: LIMFUNC2:th 83
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) = 0 &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0) &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1 - b3,b1.[
holds 0 <= b2 . b4))
holds b2 ^ is_left_divergent_to+infty_in b1;
:: LIMFUNC2:th 84
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_left_convergent_in b1 &
lim_left(b2,b1) = 0 &
(for b3 being Element of REAL
st b3 < b1
holds ex b4 being Element of REAL st
b3 < b4 & b4 < b1 & b4 in dom b2 & b2 . b4 <> 0) &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1 - b3,b1.[
holds b2 . b4 <= 0))
holds b2 ^ is_left_divergent_to-infty_in b1;
:: LIMFUNC2:th 85
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) = 0 &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0) &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1,b1 + b3.[
holds 0 <= b2 . b4))
holds b2 ^ is_right_divergent_to+infty_in b1;
:: LIMFUNC2:th 86
theorem
for b1 being Element of REAL
for b2 being Function-like Relation of REAL,REAL
st b2 is_right_convergent_in b1 &
lim_right(b2,b1) = 0 &
(for b3 being Element of REAL
st b1 < b3
holds ex b4 being Element of REAL st
b4 < b3 & b1 < b4 & b4 in dom b2 & b2 . b4 <> 0) &
(ex b3 being Element of REAL st
0 < b3 &
(for b4 being Element of REAL
st b4 in (dom b2) /\ ].b1,b1 + b3.[
holds b2 . b4 <= 0))
holds b2 ^ is_right_divergent_to-infty_in b1;