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, SEQ_2, ORDINAL2, FUNCT_1, LIMFUNC1, FUNCT_2, XXREAL_1, COMPLEX1, XXREAL_2, NAT_1, VALUED_0, ORDINAL4, LIMFUNC2, ASYMPT_1;
notations HIDDEN, TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, COMPLEX1, REAL_1, NAT_1, FUNCT_1, FUNCT_2, FUNCOP_1, VALUED_0, VALUED_1, SEQ_1, COMSEQ_2, SEQ_2, RELSET_1, RCOMP_1, PARTFUN1, RFUNCT_1, LIMFUNC1, XXREAL_0;
definitions ;
theorems TARSKI, NAT_1, ABSVALUE, SEQ_1, SEQ_2, SEQM_3, SEQ_4, RFUNCT_1, RFUNCT_2, LIMFUNC1, RCOMP_1, FUNCT_1, XREAL_0, XBOOLE_0, XBOOLE_1, XCMPLX_0, XREAL_1, COMPLEX1, XXREAL_0, VALUED_1, XXREAL_1, FUNCT_2, VALUED_0, ORDINAL1;
schemes SEQ_1, FUNCT_2;
registrations ORDINAL1, RELSET_1, NUMBERS, XREAL_0, NAT_1, MEMBERED, XBOOLE_0, VALUED_0, VALUED_1, FUNCT_2, SEQ_4, SEQ_1, SEQ_2;
constructors HIDDEN, FUNCOP_1, REAL_1, NAT_1, COMPLEX1, SEQ_2, SEQM_3, PROB_1, RCOMP_1, RFUNCT_1, RFUNCT_2, LIMFUNC1, PARTFUN1, VALUED_1, RELSET_1, BINOP_2, RVSUM_1, COMSEQ_2, SEQ_1;
requirements HIDDEN, REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
equalities LIMFUNC1, PROB_1, VALUED_1;
expansions LIMFUNC1;
Lm1:
for r, g, 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 Subset of REAL st rng seq c= (dom (f1 (#) f2)) /\ X holds
( rng seq c= dom (f1 (#) f2) & rng seq c= X & 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 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 Subset of REAL st rng seq c= (dom (f1 + f2)) /\ X holds
( rng seq c= dom (f1 + f2) & rng seq c= X & dom (f1 + f2) = (dom f1) /\ (dom f2) & rng seq c= (dom f1) /\ X & rng seq c= (dom f2) /\ X )
Lm5:
for x0 being Real
for seq being Real_Sequence
for f being PartFunc of REAL,REAL st ( for g1 being Real ex r being Real st
( x0 < r & ( for r1 being Real st r1 < r & x0 < r1 & r1 in dom f holds
f . r1 < g1 ) ) ) & seq is convergent & lim seq = x0 & rng seq c= (dom f) /\ (right_open_halfline x0) holds
f /* seq is divergent_to-infty
theorem Th63:
for
x0 being
Real for
f,
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_left_convergent_in x0 &
f2 is_left_convergent_in x0 &
lim_left (
f1,
x0)
= lim_left (
f2,
x0) & ( for
r being
Real st
r < x0 holds
ex
g being
Real st
(
r < g &
g < x0 &
g in dom f ) ) & ex
r being
Real st
(
0 < r & ( for
g being
Real st
g in (dom f) /\ ].(x0 - r),x0.[ holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) & ( (
(dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ &
(dom f) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ ) or (
(dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ &
(dom f) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ ) ) ) holds
(
f is_left_convergent_in x0 &
lim_left (
f,
x0)
= lim_left (
f1,
x0) )
theorem Th65:
for
x0 being
Real for
f,
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_right_convergent_in x0 &
f2 is_right_convergent_in x0 &
lim_right (
f1,
x0)
= lim_right (
f2,
x0) & ( for
r being
Real st
x0 < r holds
ex
g being
Real st
(
g < r &
x0 < g &
g in dom f ) ) & ex
r being
Real st
(
0 < r & ( for
g being
Real st
g in (dom f) /\ ].x0,(x0 + r).[ holds
(
f1 . g <= f . g &
f . g <= f2 . g ) ) & ( (
(dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ &
(dom f) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ ) or (
(dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ &
(dom f) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ ) ) ) holds
(
f is_right_convergent_in x0 &
lim_right (
f,
x0)
= lim_right (
f1,
x0) )
theorem
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_left_convergent_in x0 &
f2 is_left_convergent_in x0 & ex
r being
Real st
(
0 < r & ( (
(dom f1) /\ ].(x0 - r),x0.[ c= (dom f2) /\ ].(x0 - r),x0.[ & ( for
g being
Real st
g in (dom f1) /\ ].(x0 - r),x0.[ holds
f1 . g <= f2 . g ) ) or (
(dom f2) /\ ].(x0 - r),x0.[ c= (dom f1) /\ ].(x0 - r),x0.[ & ( for
g being
Real st
g in (dom f2) /\ ].(x0 - r),x0.[ holds
f1 . g <= f2 . g ) ) ) ) holds
lim_left (
f1,
x0)
<= lim_left (
f2,
x0)
theorem
for
x0 being
Real for
f1,
f2 being
PartFunc of
REAL,
REAL st
f1 is_right_convergent_in x0 &
f2 is_right_convergent_in x0 & ex
r being
Real st
(
0 < r & ( (
(dom f1) /\ ].x0,(x0 + r).[ c= (dom f2) /\ ].x0,(x0 + r).[ & ( for
g being
Real st
g in (dom f1) /\ ].x0,(x0 + r).[ holds
f1 . g <= f2 . g ) ) or (
(dom f2) /\ ].x0,(x0 + r).[ c= (dom f1) /\ ].x0,(x0 + r).[ & ( for
g being
Real st
g in (dom f2) /\ ].x0,(x0 + r).[ holds
f1 . g <= f2 . g ) ) ) ) holds
lim_right (
f1,
x0)
<= lim_right (
f2,
x0)