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, Def2; :: 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, Def2; :: 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, Def2;
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 2; :: 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 5; :: 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: for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom f ) by A12, LIMFUNC2:def 5;
A14: 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
A15: s is convergent and
A16: lim s = x0 and
A17: 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
A18: for n being Element of NAT st k <= n holds
s . n < x0 ;
A19: rng s c= dom f by A17, XBOOLE_1:1;
A20: 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
A21: (s ^\ k) . n = x by FUNCT_2:190;
s . (n + k) < x0 by A18, 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 A22: x in left_open_halfline x0 by A21, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A21, NAT_1:def 3;
hence x in (dom f) /\ (left_open_halfline x0) by A19, A22, XBOOLE_0:def 4; :: thesis: verum
end;
A23: f /* (s ^\ k) = (f /* s) ^\ k by A17, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A15, A16, SEQ_4:33;
then f /* (s ^\ k) is divergent_to+infty by A11, A15, A20, LIMFUNC2:def 2;
hence f /* s is divergent_to+infty by A23, LIMFUNC1:34; :: thesis: verum
end;
suppose A24: 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
A25: for n being Element of NAT st k <= n holds
s . n > x0 ;
A26: rng s c= dom f by A17, XBOOLE_1:1;
A27: 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
A28: (s ^\ k) . n = x by FUNCT_2:190;
x0 < s . (n + k) by A25, 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 A29: x in right_open_halfline x0 by A28, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A28, NAT_1:def 3;
hence x in (dom f) /\ (right_open_halfline x0) by A26, A29, XBOOLE_0:def 4; :: thesis: verum
end;
A30: f /* (s ^\ k) = (f /* s) ^\ k by A17, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A15, A16, SEQ_4:33;
then f /* (s ^\ k) is divergent_to+infty by A12, A15, A27, LIMFUNC2:def 5;
hence f /* s is divergent_to+infty by A30, LIMFUNC1:34; :: thesis: verum
end;
suppose A31: 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[ Nat] means s . $1 < x0;
A32: 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
A33: k <= n and
A34: s . n <= x0 by A31;
take n = n; :: thesis: ( k <= n & s . n < x0 )
thus k <= n by A33; :: thesis: s . n < x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A17, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n < x0 by A34, XXREAL_0:1; :: thesis: verum
end;
then ex m1 being Element of NAT st
( 0 <= m1 & s . m1 < x0 ) ;
then A35: ex m being Nat st S1[m] ;
consider M being Nat such that
A36: ( S1[M] & ( for n being Nat st S1[n] holds
M <= n ) ) from NAT_1:sch 5(A35);
defpred S2[ Nat] means s . $1 > x0;
defpred S3[ 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 S4[ Element of NAT , set , set ] means S3[$2,$3];
reconsider M9 = M as Element of NAT by ORDINAL1:def 13;
A37: 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
A38: n + 1 <= m and
A39: s . m < x0 by A32;
take m = m; :: thesis: ( n < m & s . m < x0 )
thus ( n < m & s . m < x0 ) by A38, A39, NAT_1:13; :: thesis: verum
end;
A40: for n, x being Element of NAT ex y being Element of NAT st S4[n,x,y]
proof
let n, x be Element of NAT ; :: thesis: ex y being Element of NAT st S4[n,x,y]
defpred S5[ Nat] means ( x < $1 & s . $1 < x0 );
ex m being Element of NAT st S5[m] by A37;
then A41: ex m being Nat st S5[m] ;
consider l being Nat such that
A42: ( S5[l] & ( for k being Nat st S5[k] holds
l <= k ) ) from NAT_1:sch 5(A41);
 take l ; :: thesis: ( l is Element of REAL & l is Element of NAT & S4[n,x,l] )
l in NAT by ORDINAL1:def 13;
hence ( l is Element of REAL & l is Element of NAT & S4[n,x,l] ) by A42; :: thesis: verum
end;
consider F being Function of NAT ,NAT such that
A43: ( F . 0 = M9 & ( for n being Element of NAT holds S4[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch 2(A40);
A44: rng F c= NAT by RELAT_1:def 19;
then A45: rng F c= REAL by XBOOLE_1:1;
A46: dom F = NAT by FUNCT_2:def 1;
then reconsider F = F as Real_Sequence by A45, RELSET_1:11;
A47: now
let n be Element of NAT ; :: thesis: F . n is Element of NAT
F . n in rng F by A46, FUNCT_1:def 5;
hence F . n is Element of NAT by A44; :: thesis: verum
end;
now
let n be Element of NAT ; :: thesis: F . n < F . (n + 1)
A48: F . (n + 1) is Element of NAT by A47;
F . n is Element of NAT by A47;
hence F . n < F . (n + 1) by A43, A48; :: thesis: verum
end;
then reconsider F = F as V33() sequence of NAT by SEQM_3:def 11;
A49: s * F is subsequence of s by VALUED_0:def 17;
then rng (s * F) c= rng s by VALUED_0:21;
then A50: rng (s * F) c= (dom f) \ {x0} by A17, XBOOLE_1:1;
A51: for n being Element of NAT st s . n < x0 holds
ex m being Element of NAT st F . m = n
proof
defpred S5[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) );
assume ex n being Element of NAT st S5[n] ; :: thesis: contradiction
then A52: ex n being Nat st S5[n] ;
consider M1 being Nat such that
A53: ( S5[M1] & ( for n being Nat st S5[n] holds
M1 <= n ) ) from NAT_1:sch 5(A52);
 defpred S6[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 );
A54: ex n being Nat st S6[n]
proof
take M ; :: thesis: S6[M]
A55: M <> M1 by A43, A53;
M <= M1 by A36, A53;
hence M < M1 by A55, XXREAL_0:1; :: thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M )
thus s . M < x0 by A36; :: thesis: ex m being Element of NAT st F . m = M
take 0 ; :: thesis: F . 0 = M
thus F . 0 = M by A43; :: thesis: verum
end;
A56: for n being Nat st S6[n] holds
n <= M1 ;
consider MX being Nat such that
A57: ( S6[MX] & ( for n being Nat st S6[n] holds
n <= MX ) ) from NAT_1:sch 6(A56, A54);
 A58: for k being Element of NAT st MX < k & k < M1 holds
s . k >= x0
proof
given k being Element of NAT such that A59: MX < k and
A60: k < M1 and
A61: 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
A62: F . m = MX by A57;
M1 in NAT by ORDINAL1:def 13;
then A63: F . (m + 1) <= M1 by A43, A53, A57, A62;
A64: s . (F . (m + 1)) < x0 by A43, A62;
A65: MX < F . (m + 1) by A43, A62;
now
assume F . (m + 1) <> M1 ; :: thesis: contradiction
then F . (m + 1) < M1 by A63, XXREAL_0:1;
hence contradiction by A58, A65, A64; :: thesis: verum
end;
hence contradiction by A53; :: thesis: verum
end;
A66: 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
A67: k <= n and
A68: s . n >= x0 by A24;
take n = n; :: thesis: ( k <= n & s . n > x0 )
thus k <= n by A67; :: thesis: s . n > x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A17, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n > x0 by A68, XXREAL_0:1; :: thesis: verum
end;
then ex mn being Element of NAT st
( 0 <= mn & s . mn > x0 ) ;
then A69: ex m being Nat st S2[m] ;
consider N being Nat such that
A70: ( S2[N] & ( for n being Nat st S2[n] holds
N <= n ) ) from NAT_1:sch 5(A69);
A71: for n being Element of NAT holds (s * F) . n < x0
proof
defpred S5[ Element of NAT ] means (s * F) . $1 < x0;
A72: for k being Element of NAT st S5[k] holds
S5[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S5[k] implies S5[k + 1] )
assume (s * F) . k < x0 ; :: thesis: S5[k + 1]
S3[F . k,F . (k + 1)] by A43;
then s . (F . (k + 1)) < x0 ;
hence S5[k + 1] by FUNCT_2:21; :: thesis: verum
end;
A73: S5[ 0 ] by A36, A43, FUNCT_2:21;
thus for k being Element of NAT holds S5[k] from NAT_1:sch 1(A73, A72); :: thesis: verum
end;
A74: 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 A75: x in rng (s * F) ; :: thesis: x in (dom f) /\ (left_open_halfline x0)
then consider n being Element of NAT such that
A76: (s * F) . n = x by FUNCT_2:190;
(s * F) . n < x0 by A71;
then x in { g1 where g1 is Real : g1 < x0 } by A76;
then A77: x in left_open_halfline x0 by XXREAL_1:229;
x in dom f by A50, A75, XBOOLE_0:def 5;
hence x in (dom f) /\ (left_open_halfline x0) by A77, XBOOLE_0:def 4; :: thesis: verum
end;
defpred S5[ 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 S6[ Element of NAT , set , set ] means S5[$2,$3];
A78: s * F is convergent by A15, A49, SEQ_4:29;
reconsider N9 = N as Element of NAT by ORDINAL1:def 13;
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 A66;
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 S6[n,x,y]
proof
let n, x be Element of NAT ; :: thesis: ex y being Element of NAT st S6[n,x,y]
defpred S7[ Nat] means ( x < $1 & s . $1 > x0 );
ex m being Element of NAT st S7[m] by A79;
then A83: ex m being Nat st S7[m] ;
consider l being Nat such that
A84: ( S7[l] & ( for k being Nat st S7[k] holds
l <= k ) ) from NAT_1:sch 5(A83);
 reconsider l = l as Element of NAT by ORDINAL1:def 13;
take l ; :: thesis: S6[n,x,l]
thus S6[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 S6[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_1:11;
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 5;
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 11;
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 A17, XBOOLE_1:1;
defpred S7[ 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 S7[n] ; :: thesis: contradiction
then A94: ex n being Nat st S7[n] ;
consider N1 being Nat such that
A95: ( S7[N1] & ( for n being Nat st S7[n] holds
N1 <= n ) ) from NAT_1:sch 5(A94);
 defpred S8[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 );
A96: ex n being Nat st S8[n]
proof
take N ; :: thesis: S8[N]
A97: N <> N1 by A85, A95;
N <= N1 by A70, 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 A70; :: 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 S8[n] holds
n <= N1 ;
consider NX being Nat such that
A99: ( S8[NX] & ( for n being Nat st S8[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 13;
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 S8[ Element of NAT ] means (s * G) . $1 > x0;
A109: for k being Element of NAT st S8[k] holds
S8[k + 1]
proof
let k be Element of NAT ; :: thesis: ( S8[k] implies S8[k + 1] )
assume (s * G) . k > x0 ; :: thesis: S8[k + 1]
S5[G . k,G . (k + 1)] by A85;
then s . (G . (k + 1)) > x0 ;
hence S8[k + 1] by FUNCT_2:21; :: thesis: verum
end;
A110: S8[ 0 ] by A70, A85, FUNCT_2:21;
thus for k being Element of NAT holds S8[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_2:190;
(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 A15, A91, SEQ_4:29;
lim (s * G) = x0 by A15, A16, A91, SEQ_4:30;
then A116: f /* (s * G) is divergent_to+infty by A12, A115, A111, LIMFUNC2:def 5;
lim (s * F) = x0 by A15, A16, A49, SEQ_4:30;
then A117: f /* (s * F) is divergent_to+infty by A11, A78, A74, LIMFUNC2:def 2;
now
let r be Real; :: thesis: ex n being Element of NAT st
for k being Element of NAT st n <= k holds
r < (f /* s) . k
consider n1 being Element of NAT such that
A118: for k being Element of NAT st n1 <= k holds
r < (f /* (s * F)) . k by A117, LIMFUNC1:def 4;
consider n2 being Element of NAT such that
A119: for k being Element of NAT st n2 <= k holds
r < (f /* (s * G)) . k by A116, LIMFUNC1:def 4;
take n = max (F . n1),(G . n2); :: thesis: for k being Element of NAT st n <= k holds
r < (f /* s) . k
let k be Element of NAT ; :: thesis: ( n <= k implies r < (f /* s) . k )
assume A120: n <= k ; :: thesis: r < (f /* s) . k
s . k in rng s by VALUED_0:28;
then not s . k in {x0} by A17, XBOOLE_0:def 5;
then A121: s . k <> x0 by TARSKI:def 1;
now
per cases ( s . k < x0 or s . k > x0 ) by A121, XXREAL_0:1;
suppose s . k < x0 ; :: thesis: r < (f /* s) . k
then consider l being Element of NAT such that
A122: k = F . l by A51;
F . n1 <= n by XXREAL_0:25;
then F . n1 <= k by A120, XXREAL_0:2;
then l >= n1 by A122, SEQM_3:7;
then r < (f /* (s * F)) . l by A118;
then r < f . ((s * F) . l) by A50, FUNCT_2:185, XBOOLE_1:1;
then r < f . (s . k) by A122, FUNCT_2:21;
hence r < (f /* s) . k by A17, FUNCT_2:185, XBOOLE_1:1; :: thesis: verum
end;
suppose s . k > x0 ; :: thesis: r < (f /* s) . k
then consider l being Element of NAT such that
A123: k = G . l by A93;
G . n2 <= n by XXREAL_0:25;
then G . n2 <= k by A120, XXREAL_0:2;
then l >= n2 by A123, SEQM_3:7;
then r < (f /* (s * G)) . l by A119;
then r < f . ((s * G) . l) by A92, FUNCT_2:185, XBOOLE_1:1;
then r < f . (s . k) by A123, FUNCT_2:21;
hence r < (f /* s) . k by A17, FUNCT_2:185, XBOOLE_1:1; :: thesis: verum
end;
end;
end;
hence r < (f /* s) . k ; :: thesis: verum
end;
hence f /* s is divergent_to+infty by LIMFUNC1:def 4; :: 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;
for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) by A11, LIMFUNC2:def 2;
then 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 A13, Th8;
hence f is_divergent_to+infty_in x0 by A14, Def2; :: thesis: verum