let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL holds
( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) )
thus ( f is_divergent_to-infty_in x0 implies ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) :: thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 implies f is_divergent_to-infty_in x0 )
proof
assume A1: f is_divergent_to-infty_in x0 ; :: thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 )
A2: now
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) implies f /* s is divergent_to-infty )
assume that
A3: s is convergent and
A4: lim s = x0 and
A5: rng s c= (dom f) /\ (left_open_halfline x0) ; :: thesis: f /* s is divergent_to-infty
rng s c= (dom f) \ {x0} by A5, Th1;
hence f /* s is divergent_to-infty by A1, A3, A4, Def3; :: thesis: verum
end;
A6: now
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) implies f /* s is divergent_to-infty )
assume that
A7: s is convergent and
A8: lim s = x0 and
A9: rng s c= (dom f) /\ (right_open_halfline x0) ; :: thesis: f /* s is divergent_to-infty
rng s c= (dom f) \ {x0} by A9, Th1;
hence f /* s is divergent_to-infty by A1, A7, A8, Def3; :: thesis: verum
end;
A10: 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 ) by A1, Def3;
then for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) by Th8;
hence f is_left_divergent_to-infty_in x0 by A2, LIMFUNC2:def 3; :: thesis: f is_right_divergent_to-infty_in x0
for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) by A10, Th8;
hence f is_right_divergent_to-infty_in x0 by A6, LIMFUNC2:def 6; :: thesis: verum
end;
assume that
A11: f is_left_divergent_to-infty_in x0 and
A12: f is_right_divergent_to-infty_in x0 ; :: thesis: f is_divergent_to-infty_in x0
A13: now
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty )
assume that
A14: s is convergent and
A15: lim s = x0 and
A16: rng s c= (dom f) \ {x0} ; :: thesis: f /* s is divergent_to-infty
now
per cases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
s . n < x0 or for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n >= x0 ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
s . n < x0 ; :: thesis: f /* s is divergent_to-infty
then consider k being Element of NAT such that
A17: for n being Element of NAT st k <= n holds
s . n < x0 ;
A18: rng s c= dom f by A16, XBOOLE_1:1;
A19: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (left_open_halfline x0) )
assume x in rng (s ^\ k) ; :: thesis: x in (dom f) /\ (left_open_halfline x0)
then consider n being Element of NAT such that
A20: (s ^\ k) . n = x by FUNCT_2:113;
s . (n + k) < x0 by A17, NAT_1:12;
then s . (n + k) in { g1 where g1 is Real : g1 < x0 } ;
then s . (n + k) in left_open_halfline x0 by XXREAL_1:229;
then A21: x in left_open_halfline x0 by A20, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A20, NAT_1:def 3;
hence x in (dom f) /\ (left_open_halfline x0) by A18, A21, XBOOLE_0:def 4; :: thesis: verum
end;
A22: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A14, A15, SEQ_4:20;
then f /* (s ^\ k) is divergent_to-infty by A11, A14, A19, LIMFUNC2:def 3;
hence f /* s is divergent_to-infty by A22, LIMFUNC1:7; :: thesis: verum
end;
suppose A23: for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n >= x0 ) ; :: thesis: f /* s is divergent_to-infty
now
per cases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
x0 < s . n or for k being Element of NAT ex n being Element of NAT st
( k <= n & x0 >= s . n ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
x0 < s . n ; :: thesis: f /* s is divergent_to-infty
then consider k being Element of NAT such that
A24: for n being Element of NAT st k <= n holds
s . n > x0 ;
A25: rng s c= dom f by A16, XBOOLE_1:1;
A26: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (right_open_halfline x0) )
assume x in rng (s ^\ k) ; :: thesis: x in (dom f) /\ (right_open_halfline x0)
then consider n being Element of NAT such that
A27: (s ^\ k) . n = x by FUNCT_2:113;
x0 < s . (n + k) by A24, NAT_1:12;
then s . (n + k) in { g1 where g1 is Real : x0 < g1 } ;
then s . (n + k) in right_open_halfline x0 by XXREAL_1:230;
then A28: x in right_open_halfline x0 by A27, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A27, NAT_1:def 3;
hence x in (dom f) /\ (right_open_halfline x0) by A25, A28, XBOOLE_0:def 4; :: thesis: verum
end;
A29: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A14, A15, SEQ_4:20;
then f /* (s ^\ k) is divergent_to-infty by A12, A14, A26, LIMFUNC2:def 6;
hence f /* s is divergent_to-infty by A29, LIMFUNC1:7; :: thesis: verum
end;
suppose A30: for k being Element of NAT ex n being Element of NAT st
( k <= n & x0 >= s . n ) ; :: thesis: f /* s is divergent_to-infty
defpred S1[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds
( n < m & s . m < x0 & ( for k being Element of NAT st n < k & s . k < x0 holds
m <= k ) );
defpred S2[ Element of NAT , set , set ] means S1[$2,$3];
defpred S3[ Nat] means s . $1 < x0;
A31: now
let k be Element of NAT ; :: thesis: ex n being Element of NAT st
( k <= n & s . n < x0 )
consider n being Element of NAT such that
A32: k <= n and
A33: s . n <= x0 by A30;
take n = n; :: thesis: ( k <= n & s . n < x0 )
thus k <= n by A32; :: thesis: s . n < x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A16, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n < x0 by A33, XXREAL_0:1; :: thesis: verum
end;
then ex m1 being Element of NAT st
( 0 <= m1 & s . m1 < x0 ) ;
then A34: ex m being Nat st S3[m] ;
consider M being Nat such that
A35: ( S3[M] & ( for n being Nat st S3[n] holds
M <= n ) ) from NAT_1:sch 5(A34);
reconsider M9 = M as Element of NAT by ORDINAL1:def 12;
A36: now
let n be Element of NAT ; :: thesis: ex m being Element of NAT st
( n < m & s . m < x0 )
consider m being Element of NAT such that
A37: n + 1 <= m and
A38: s . m < x0 by A31;
take m = m; :: thesis: ( n < m & s . m < x0 )
thus ( n < m & s . m < x0 ) by A37, A38, NAT_1:13; :: thesis: verum
end;
A39: for n, x being Element of NAT ex y being Element of NAT st S2[n,x,y]
proof
let n, x be Element of NAT ; :: thesis: ex y being Element of NAT st S2[n,x,y]
defpred S4[ Nat] means ( x < $1 & s . $1 < x0 );
ex m being Element of NAT st S4[m] by A36;
then A40: ex m being Nat st S4[m] ;
consider l being Nat such that
A41: ( S4[l] & ( for k being Nat st S4[k] holds
l <= k ) ) from NAT_1:sch 5(A40);
 take l ; :: thesis: ( l is Element of REAL & l is Element of NAT & S2[n,x,l] )
l in NAT by ORDINAL1:def 12;
hence ( l is Element of REAL & l is Element of NAT & S2[n,x,l] ) by A41; :: thesis: verum
end;
consider F being Function of NAT,NAT such that
A42: ( F . 0 = M9 & ( for n being Element of NAT holds S2[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch 2(A39);
A43: rng F c= NAT by RELAT_1:def 19;
then A44: rng F c= REAL by XBOOLE_1:1;
A45: dom F = NAT by FUNCT_2:def 1;
then reconsider F = F as Real_Sequence by A44, RELSET_1:4;
A46: now
let n be Element of NAT ; :: thesis: F . n is Element of NAT
F . n in rng F by A45, FUNCT_1:def 3;
hence F . n is Element of NAT by A43; :: thesis: verum
end;
now
let n be Element of NAT ; :: thesis: F . n < F . (n + 1)
A47: F . (n + 1) is Element of NAT by A46;
F . n is Element of NAT by A46;
hence F . n < F . (n + 1) by A42, A47; :: thesis: verum
end;
then reconsider F = F as V33() sequence of NAT by SEQM_3:def 6;
A48: s * F is subsequence of s by VALUED_0:def 17;
then rng (s * F) c= rng s by VALUED_0:21;
then A49: rng (s * F) c= (dom f) \ {x0} by A16, XBOOLE_1:1;
defpred S4[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) );
A50: for n being Element of NAT st s . n < x0 holds
ex m being Element of NAT st F . m = n
proof
assume ex n being Element of NAT st S4[n] ; :: thesis: contradiction
then A51: ex n being Nat st S4[n] ;
consider M1 being Nat such that
A52: ( S4[M1] & ( for n being Nat st S4[n] holds
M1 <= n ) ) from NAT_1:sch 5(A51);
 defpred S5[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 );
A53: ex n being Nat st S5[n]
proof
take M ; :: thesis: S5[M]
A54: M <> M1 by A42, A52;
M <= M1 by A35, A52;
hence M < M1 by A54, XXREAL_0:1; :: thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M )
thus s . M < x0 by A35; :: thesis: ex m being Element of NAT st F . m = M
take 0 ; :: thesis: F . 0 = M
thus F . 0 = M by A42; :: thesis: verum
end;
A55: for n being Nat st S5[n] holds
n <= M1 ;
consider MX being Nat such that
A56: ( S5[MX] & ( for n being Nat st S5[n] holds
n <= MX ) ) from NAT_1:sch 6(A55, A53);
 A57: for k being Element of NAT st MX < k & k < M1 holds
s . k >= x0
proof
given k being Element of NAT such that A58: MX < k and
A59: k < M1 and
A60: s . k < x0 ; :: thesis: contradiction
now
per cases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ;
end;
end;
hence contradiction ; :: thesis: verum
end;
consider m being Element of NAT such that
A61: F . m = MX by A56;
M1 in NAT by ORDINAL1:def 12;
then A62: F . (m + 1) <= M1 by A42, A52, A56, A61;
A63: s . (F . (m + 1)) < x0 by A42, A61;
A64: MX < F . (m + 1) by A42, A61;
now
assume F . (m + 1) <> M1 ; :: thesis: contradiction
then F . (m + 1) < M1 by A62, XXREAL_0:1;
hence contradiction by A57, A64, A63; :: thesis: verum
end;
hence contradiction by A52; :: thesis: verum
end;
defpred S5[ Nat] means s . $1 > x0;
A65: now
let k be Element of NAT ; :: thesis: ex n being Element of NAT st
( k <= n & s . n > x0 )
consider n being Element of NAT such that
A66: k <= n and
A67: s . n >= x0 by A23;
take n = n; :: thesis: ( k <= n & s . n > x0 )
thus k <= n by A66; :: thesis: s . n > x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A16, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n > x0 by A67, XXREAL_0:1; :: thesis: verum
end;
then ex mn being Element of NAT st
( 0 <= mn & s . mn > x0 ) ;
then A68: ex m being Nat st S5[m] ;
consider N being Nat such that
A69: ( S5[N] & ( for n being Nat st S5[n] holds
N <= n ) ) from NAT_1:sch 5(A68);
A70: for n being Element of NAT holds (s * F) . n < x0
proof
defpred S6[ Element of NAT ] means (s * F) . $1 < x0;
A71: for k being Element of NAT st S6[k] holds
S6[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S6[k] implies S6[k + 1] )
assume (s * F) . k < x0 ; :: thesis: S6[k + 1]
S1[F . k,F . (k + 1)] by A42;
then s . (F . (k + 1)) < x0 ;
hence S6[k + 1] by FUNCT_2:15; :: thesis: verum
end;
A72: S6[ 0 ] by A35, A42, FUNCT_2:15;
thus for k being Element of NAT holds S6[k] from NAT_1:sch 1(A72, A71); :: thesis: verum
end;
A73: rng (s * F) c= (dom f) /\ (left_open_halfline x0)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s * F) or x in (dom f) /\ (left_open_halfline x0) )
assume A74: x in rng (s * F) ; :: thesis: x in (dom f) /\ (left_open_halfline x0)
then consider n being Element of NAT such that
A75: (s * F) . n = x by FUNCT_2:113;
(s * F) . n < x0 by A70;
then x in { g1 where g1 is Real : g1 < x0 } by A75;
then A76: x in left_open_halfline x0 by XXREAL_1:229;
x in dom f by A49, A74, XBOOLE_0:def 5;
hence x in (dom f) /\ (left_open_halfline x0) by A76, XBOOLE_0:def 4; :: thesis: verum
end;
defpred S6[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds
( n < m & s . m > x0 & ( for k being Element of NAT st n < k & s . k > x0 holds
m <= k ) );
defpred S7[ Element of NAT , set , set ] means S6[$2,$3];
A77: s * F is convergent by A14, A48, SEQ_4:16;
lim (s * F) = x0 by A14, A15, A48, SEQ_4:17;
then A78: f /* (s * F) is divergent_to-infty by A11, A77, A73, LIMFUNC2:def 3;
reconsider N9 = N as Element of NAT by ORDINAL1:def 12;
A79: now
let n be Element of NAT ; :: thesis: ex m being Element of NAT st
( n < m & s . m > x0 )
consider m being Element of NAT such that
A80: n + 1 <= m and
A81: s . m > x0 by A65;
take m = m; :: thesis: ( n < m & s . m > x0 )
thus ( n < m & s . m > x0 ) by A80, A81, NAT_1:13; :: thesis: verum
end;
A82: for n, x being Element of NAT ex y being Element of NAT st S7[n,x,y]
proof
let n, x be Element of NAT ; :: thesis: ex y being Element of NAT st S7[n,x,y]
defpred S8[ Nat] means ( x < $1 & s . $1 > x0 );
ex m being Element of NAT st S8[m] by A79;
then A83: ex m being Nat st S8[m] ;
consider l being Nat such that
A84: ( S8[l] & ( for k being Nat st S8[k] holds
l <= k ) ) from NAT_1:sch 5(A83);
 take l ; :: thesis: ( l is Element of REAL & l is Element of NAT & S7[n,x,l] )
l in NAT by ORDINAL1:def 12;
hence ( l is Element of REAL & l is Element of NAT & S7[n,x,l] ) by A84; :: thesis: verum
end;
consider G being Function of NAT,NAT such that
A85: ( G . 0 = N9 & ( for n being Element of NAT holds S7[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch 2(A82);
A86: rng G c= NAT by RELAT_1:def 19;
then A87: rng G c= REAL by XBOOLE_1:1;
A88: dom G = NAT by FUNCT_2:def 1;
then reconsider G = G as Real_Sequence by A87, RELSET_1:4;
A89: now
let n be Element of NAT ; :: thesis: G . n is Element of NAT
G . n in rng G by A88, FUNCT_1:def 3;
hence G . n is Element of NAT by A86; :: thesis: verum
end;
now
let n be Element of NAT ; :: thesis: G . n < G . (n + 1)
A90: G . (n + 1) is Element of NAT by A89;
G . n is Element of NAT by A89;
hence G . n < G . (n + 1) by A85, A90; :: thesis: verum
end;
then reconsider G = G as V33() sequence of NAT by SEQM_3:def 6;
A91: s * G is subsequence of s by VALUED_0:def 17;
then rng (s * G) c= rng s by VALUED_0:21;
then A92: rng (s * G) c= (dom f) \ {x0} by A16, XBOOLE_1:1;
defpred S8[ Nat] means ( s . $1 > x0 & ( for m being Element of NAT holds G . m <> $1 ) );
A93: for n being Element of NAT st s . n > x0 holds
ex m being Element of NAT st G . m = n
proof
assume ex n being Element of NAT st S8[n] ; :: thesis: contradiction
then A94: ex n being Nat st S8[n] ;
consider N1 being Nat such that
A95: ( S8[N1] & ( for n being Nat st S8[n] holds
N1 <= n ) ) from NAT_1:sch 5(A94);
 defpred S9[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 );
A96: ex n being Nat st S9[n]
proof
take N ; :: thesis: S9[N]
A97: N <> N1 by A85, A95;
N <= N1 by A69, A95;
hence N < N1 by A97, XXREAL_0:1; :: thesis: ( s . N > x0 & ex m being Element of NAT st G . m = N )
thus s . N > x0 by A69; :: thesis: ex m being Element of NAT st G . m = N
take 0 ; :: thesis: G . 0 = N
thus G . 0 = N by A85; :: thesis: verum
end;
A98: for n being Nat st S9[n] holds
n <= N1 ;
consider NX being Nat such that
A99: ( S9[NX] & ( for n being Nat st S9[n] holds
n <= NX ) ) from NAT_1:sch 6(A98, A96);
 A100: for k being Element of NAT st NX < k & k < N1 holds
s . k <= x0
proof
given k being Element of NAT such that A101: NX < k and
A102: k < N1 and
A103: s . k > x0 ; :: thesis: contradiction
now
per cases ( ex m being Element of NAT st G . m = k or for m being Element of NAT holds G . m <> k ) ;
end;
end;
hence contradiction ; :: thesis: verum
end;
consider m being Element of NAT such that
A104: G . m = NX by A99;
N1 in NAT by ORDINAL1:def 12;
then A105: G . (m + 1) <= N1 by A85, A95, A99, A104;
A106: s . (G . (m + 1)) > x0 by A85, A104;
A107: NX < G . (m + 1) by A85, A104;
now
assume G . (m + 1) <> N1 ; :: thesis: contradiction
then G . (m + 1) < N1 by A105, XXREAL_0:1;
hence contradiction by A100, A107, A106; :: thesis: verum
end;
hence contradiction by A95; :: thesis: verum
end;
A108: for n being Element of NAT holds (s * G) . n > x0
proof
defpred S9[ Element of NAT ] means (s * G) . $1 > x0;
A109: for k being Element of NAT st S9[k] holds
S9[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S9[k] implies S9[k + 1] )
assume (s * G) . k > x0 ; :: thesis: S9[k + 1]
S6[G . k,G . (k + 1)] by A85;
then s . (G . (k + 1)) > x0 ;
hence S9[k + 1] by FUNCT_2:15; :: thesis: verum
end;
A110: S9[ 0 ] by A69, A85, FUNCT_2:15;
thus for k being Element of NAT holds S9[k] from NAT_1:sch 1(A110, A109); :: thesis: verum
end;
A111: rng (s * G) c= (dom f) /\ (right_open_halfline x0)
proof
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s * G) or x in (dom f) /\ (right_open_halfline x0) )
assume A112: x in rng (s * G) ; :: thesis: x in (dom f) /\ (right_open_halfline x0)
then consider n being Element of NAT such that
A113: (s * G) . n = x by FUNCT_2:113;
(s * G) . n > x0 by A108;
then x in { g1 where g1 is Real : x0 < g1 } by A113;
then A114: x in right_open_halfline x0 by XXREAL_1:230;
x in dom f by A92, A112, XBOOLE_0:def 5;
hence x in (dom f) /\ (right_open_halfline x0) by A114, XBOOLE_0:def 4; :: thesis: verum
end;
A115: s * G is convergent by A14, A91, SEQ_4:16;
lim (s * G) = x0 by A14, A15, A91, SEQ_4:17;
then A116: f /* (s * G) is divergent_to-infty by A12, A115, A111, LIMFUNC2:def 6;
now
let r be Real; :: thesis: ex n being Element of NAT st
for k being Element of NAT st n <= k holds
(f /* s) . k < r
consider n1 being Element of NAT such that
A117: for k being Element of NAT st n1 <= k holds
(f /* (s * F)) . k < r by A78, LIMFUNC1:def 5;
consider n2 being Element of NAT such that
A118: for k being Element of NAT st n2 <= k holds
(f /* (s * G)) . k < r by A116, LIMFUNC1:def 5;
take n = max ((F . n1),(G . n2)); :: thesis: for k being Element of NAT st n <= k holds
(f /* s) . k < r
let k be Element of NAT ; :: thesis: ( n <= k implies (f /* s) . k < r )
assume A119: n <= k ; :: thesis: (f /* s) . k < r
s . k in rng s by VALUED_0:28;
then not s . k in {x0} by A16, XBOOLE_0:def 5;
then A120: s . k <> x0 by TARSKI:def 1;
now
per cases ( s . k < x0 or s . k > x0 ) by A120, XXREAL_0:1;
suppose s . k < x0 ; :: thesis: (f /* s) . k < r
then consider l being Element of NAT such that
A121: k = F . l by A50;
F . n1 <= n by XXREAL_0:25;
then F . n1 <= k by A119, XXREAL_0:2;
then l >= n1 by A121, SEQM_3:1;
then (f /* (s * F)) . l < r by A117;
then f . ((s * F) . l) < r by A49, FUNCT_2:108, XBOOLE_1:1;
then f . (s . k) < r by A121, FUNCT_2:15;
hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1; :: thesis: verum
end;
suppose s . k > x0 ; :: thesis: (f /* s) . k < r
then consider l being Element of NAT such that
A122: k = G . l by A93;
G . n2 <= n by XXREAL_0:25;
then G . n2 <= k by A119, XXREAL_0:2;
then l >= n2 by A122, SEQM_3:1;
then (f /* (s * G)) . l < r by A118;
then f . ((s * G) . l) < r by A92, FUNCT_2:108, XBOOLE_1:1;
then f . (s . k) < r by A122, FUNCT_2:15;
hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1; :: thesis: verum
end;
end;
end;
hence (f /* s) . k < r ; :: thesis: verum
end;
hence f /* s is divergent_to-infty by LIMFUNC1:def 5; :: thesis: verum
end;
end;
end;
hence f /* s is divergent_to-infty ; :: thesis: verum
end;
end;
end;
hence f /* s is divergent_to-infty ; :: thesis: verum
end;
now
let r1, r2 be Real; :: thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) )
assume that
A123: r1 < x0 and
A124: x0 < r2 ; :: thesis: ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
consider g1 being Real such that
A125: r1 < g1 and
A126: g1 < x0 and
A127: g1 in dom f by A11, A123, LIMFUNC2:def 3;
consider g2 being Real such that
A128: g2 < r2 and
A129: x0 < g2 and
A130: g2 in dom f by A12, A124, LIMFUNC2:def 6;
take g1 = g1; :: thesis: ex g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
take g2 = g2; :: thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A125, A126, A127, A128, A129, A130; :: thesis: verum
end;
hence f is_divergent_to-infty_in x0 by A13, Def3; :: thesis: verum