environ
vocabularies HIDDEN, NUMBERS, REAL_1, SUBSET_1, SEQ_1, PARTFUN1, CARD_1, ARYTM_3, XXREAL_0, ARYTM_1, RELAT_1, TARSKI, VALUED_1, XBOOLE_0, LIMFUNC1, FUNCT_1, COMPLEX1, SEQ_2, ORDINAL2, XXREAL_1, FUNCT_2, LIMFUNC2, NAT_1, VALUED_0, XXREAL_2, ORDINAL4, LIMFUNC3;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, COMPLEX1, REAL_1, NAT_1, RELAT_1, FUNCT_1, FUNCT_2, VALUED_0, VALUED_1, SEQ_1, PARTFUN1, COMSEQ_2, SEQ_2, RCOMP_1, RFUNCT_1, LIMFUNC1, LIMFUNC2, RECDEF_1;
definitions TARSKI, XBOOLE_0;
theorems TARSKI, NAT_1, FUNCT_1, FUNCT_2, ABSVALUE, SEQ_1, SEQ_2, SEQM_3, SEQ_4, RFUNCT_1, RCOMP_1, RFUNCT_2, LIMFUNC1, LIMFUNC2, RELSET_1, XREAL_0, XBOOLE_0, XBOOLE_1, XCMPLX_0, XCMPLX_1, XREAL_1, COMPLEX1, XXREAL_0, ORDINAL1, VALUED_1, XXREAL_1, VALUED_0, RELAT_1, NUMBERS;
schemes NAT_1, RECDEF_1, FUNCT_2;
registrations ORDINAL1, RELSET_1, NUMBERS, XREAL_0, NAT_1, MEMBERED, VALUED_0, VALUED_1, FUNCT_2, SEQ_4;
constructors HIDDEN, REAL_1, NAT_1, COMPLEX1, SEQ_2, SEQM_3, PROB_1, RCOMP_1, PARTFUN1, RFUNCT_1, RFUNCT_2, LIMFUNC1, LIMFUNC2, VALUED_1, RECDEF_1, RELSET_1, COMSEQ_2, NUMBERS;
requirements HIDDEN, REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
equalities XBOOLE_0, VALUED_1, PROB_1, LIMFUNC1;
expansions TARSKI, LIMFUNC1;
Lm1:
for g, r, r1 being Real st 0 < g & r <= r1 holds
( r - g < r1 & r < r1 + g )
Lm2:
for seq being Real_Sequence
for f1, f2 being PartFunc of REAL,REAL
for X being set st rng seq c= (dom (f1 (#) f2)) \ X holds
( rng seq c= dom (f1 (#) f2) & dom (f1 (#) f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X )
Lm3:
for r being Real
for n being Element of NAT holds
( r - (1 / (n + 1)) < r & r < r + (1 / (n + 1)) )
Lm4:
for seq being Real_Sequence
for f1, f2 being PartFunc of REAL,REAL
for X being set st rng seq c= (dom (f1 + f2)) \ X holds
( rng seq c= dom (f1 + f2) & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= dom f1 & rng seq c= dom f2 & rng seq c= (dom f1) \ X & rng seq c= (dom f2) \ X )
theorem Th8:
for
x0 being
Real for
f being
PartFunc of
REAL,
REAL holds
( ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) iff ( ( for
r being
Real st
r < x0 holds
ex
g being
Real st
(
r < g &
g < x0 &
g in dom f ) ) & ( for
r being
Real st
x0 < r holds
ex
g being
Real st
(
g < r &
x0 < g &
g in dom f ) ) ) )
theorem Th24:
for
x0 being
Real for
f,
f1 being
PartFunc of
REAL,
REAL st
f1 is_divergent_to+infty_in x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f . g ) ) holds
f is_divergent_to+infty_in x0
theorem Th25:
for
x0 being
Real for
f,
f1 being
PartFunc of
REAL,
REAL st
f1 is_divergent_to-infty_in x0 & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f . g <= f1 . g ) ) holds
f is_divergent_to-infty_in x0
theorem Th41:
for
x0 being
Real for
f,
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 &
lim (
f1,
x0)
= lim (
f2,
x0) & ( for
r1,
r2 being
Real st
r1 < x0 &
x0 < r2 holds
ex
g1,
g2 being
Real st
(
r1 < g1 &
g1 < x0 &
g1 in dom f &
g2 < r2 &
x0 < g2 &
g2 in dom f ) ) & ex
r being
Real st
(
0 < r & ( for
g being
Real st
g in (dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) & ( (
(dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) or (
(dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) &
(dom f) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) ) ) ) holds
(
f is_convergent_in x0 &
lim (
f,
x0)
= lim (
f1,
x0) )
theorem
for
x0 being
Real for
f,
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 &
lim (
f1,
x0)
= lim (
f2,
x0) & ex
r being
Real st
(
0 < r &
].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ c= ((dom f1) /\ (dom f2)) /\ (dom f) & ( for
g being
Real st
g in ].(x0 - r),x0.[ \/ ].x0,(x0 + r).[ holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) ) holds
(
f is_convergent_in x0 &
lim (
f,
x0)
= lim (
f1,
x0) )
theorem
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_convergent_in x0 &
f2 is_convergent_in x0 & ex
r being
Real st
(
0 < r & ( (
(dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f2 . g ) ) or (
(dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) c= (dom f1) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) & ( for
g being
Real st
g in (dom f2) /\ (].(x0 - r),x0.[ \/ ].x0,(x0 + r).[) holds
f1 . g <= f2 . g ) ) ) ) holds
lim (
f1,
x0)
<= lim (
f2,
x0)