let R be Function of REAL,REAL; :: thesis: ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) )

A1: now :: thesis: ( R is RestFunc-like & ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & |.z.| < d & not |.(R . z).| / |.z.| < r ) ) ) ) implies for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) )

assume A2: R is RestFunc-like ; :: thesis: ( ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & |.z.| < d & not |.(R . z).| / |.z.| < r ) ) ) ) implies for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) )
assume ex r being Real st
( r > 0 & ( for d being Real holds
( not d > 0 or ex z being Real st
( z <> 0 & |.z.| < d & not |.(R . z).| / |.z.| < r ) ) ) ) ; :: thesis: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) )
then consider r being Real such that
A3: r > 0 and
A4: for d being Real st d > 0 holds
ex z being Real st
( z <> 0 & |.z.| < d & not |.(R . z).| / |.z.| < r ) ;
defpred S₁[ Nat, Element of REAL ] means ( $2 <> 0 & |.$2.| < 1 / ($1 + 1) & not |.(R . $2).| / |.$2.| < r );

A5: now :: thesis: for n being Element of NAT ex z being Element of REAL st S₁[n,z]

let n be Element of NAT ; :: thesis: ex z being Element of REAL st S₁[n,z]
consider z being Real such that
A6: ( z <> 0 & |.z.| < 1 / (n + 1) & not |.(R . z).| / |.z.| < r ) by A4;
reconsider z = z as Element of REAL by XREAL_0:def 1;
take z = z; :: thesis: S₁[n,z]
thus S₁[n,z] by A6; :: thesis: verum

end;

consider s being Real_Sequence such that
A7: for n being Element of NAT holds S₁[n,s . n] from FUNCT_2:sch 3(A5);
A8: for n being Nat holds S₁[n,s . n]

proof

let n be Nat; :: thesis: S₁[n,s . n]
n in NAT by ORDINAL1:def 12;
hence S₁[n,s . n] by A7; :: thesis: verum

end;

A9: now :: thesis: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - 0).| < p

let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - 0).| < p )
assume A10: 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((s . m) - 0).| < p
consider n being Nat such that
A11: p " < n by SEQ_4:3;
(p ") + 0 < n + 1 by A11, XREAL_1:8;
then A12: 1 / (n + 1) < 1 / (p ") by A10, XREAL_1:76;
take n = n; :: thesis: for m being Nat st n <= m holds
|.((s . m) - 0).| < p
let m be Nat; :: thesis: ( n <= m implies |.((s . m) - 0).| < p )
assume n <= m ; :: thesis: |.((s . m) - 0).| < p
then n + 1 <= m + 1 by XREAL_1:6;
then 1 / (m + 1) <= 1 / (n + 1) by XREAL_1:118;
then |.((s . m) - 0).| < 1 / (n + 1) by A8, XXREAL_0:2;
hence |.((s . m) - 0).| < p by A12, XXREAL_0:2; :: thesis: verum

end;

then s is convergent by SEQ_2:def 6;
then lim s = 0 by A9, SEQ_2:def 7;
then reconsider s = s as non-zero 0 -convergent Real_Sequence by A9, A8, SEQ_1:5, SEQ_2:def 6, FDIFF_1:def 1;
( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0 ) by A2, FDIFF_1:def 2;
then consider n being Nat such that
A15: for m being Nat st n <= m holds
|.((((s ") (#) (R /* s)) . m) - 0).| < r by A3, SEQ_2:def 7;
A16: n in NAT by ORDINAL1:def 12;
A18: |.(((s . n) ") * (R . (s . n))).| = |.((s . n) ").| * |.(R . (s . n)).| by COMPLEX1:65
.= |.(R . (s . n)).| / |.(s . n).| by COMPLEX1:66 ;
|.((((s ") (#) (R /* s)) . n) - 0).| = |.(((s ") . n) * ((R /* s) . n)).| by SEQ_1:8
.= |.(((s . n) ") * ((R /* s) . n)).| by VALUED_1:10
.= |.(((s . n) ") * (R . (s . n))).| by FUNCT_2:115, A16 ;
hence for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) by A8, A15, A18; :: thesis: verum

end;

now :: thesis: ( ( for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) ) implies R is RestFunc-like )

assume A19: for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) ; :: thesis: R is RestFunc-like

now :: thesis: for s being non-zero 0 -convergent Real_Sequence holds
( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0 )

let s be non-zero 0 -convergent Real_Sequence; :: thesis: ( (s ") (#) (R /* s) is convergent & lim ((s ") (#) (R /* s)) = 0 )
A20: ( s is convergent & lim s = 0 ) ;

A21: now :: thesis: for r being Real st r > 0 holds
ex n being Nat st
for m being Nat st n <= m holds
|.((((s ") (#) (R /* s)) . m) - 0).| < r

let r be Real; :: thesis: ( r > 0 implies ex n being Nat st
for m being Nat st n <= m holds
|.((((s ") (#) (R /* s)) . m) - 0).| < r )
assume A22: r > 0 ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((((s ") (#) (R /* s)) . m) - 0).| < r
consider d being Real such that
A23: d > 0 and
A24: for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r by A22, A19;
consider n being Nat such that
A25: for m being Nat st n <= m holds
|.((s . m) - 0).| < d by A20, A23, SEQ_2:def 7;
take n = n; :: thesis: for m being Nat st n <= m holds
|.((((s ") (#) (R /* s)) . m) - 0).| < r

hereby :: thesis: verum

let m be Nat; :: thesis: ( n <= m implies |.((((s ") (#) (R /* s)) . m) - 0).| < r )
A26: m in NAT by ORDINAL1:def 12;
assume n <= m ; :: thesis: |.((((s ") (#) (R /* s)) . m) - 0).| < r
then A27: |.((s . m) - 0).| < d by A25;
|.(R . (s . m)).| / |.(s . m).| = |.((s . m) ").| * |.(R . (s . m)).| by COMPLEX1:66
.= |.(((s . m) ") * (R . (s . m))).| by COMPLEX1:65
.= |.(((s . m) ") * ((R /* s) . m)).| by FUNCT_2:115, A26
.= |.(((s ") . m) * ((R /* s) . m)).| by VALUED_1:10
.= |.((((s ") (#) (R /* s)) . m) - 0).| by SEQ_1:8 ;
hence |.((((s ") (#) (R /* s)) . m) - 0).| < r by A24, A27, SEQ_1:5; :: thesis: verum

end;

end;

hence (s ") (#) (R /* s) is convergent by SEQ_2:def 6; :: thesis: lim ((s ") (#) (R /* s)) = 0
hence lim ((s ") (#) (R /* s)) = 0 by A21, SEQ_2:def 7; :: thesis: verum

end;

hence R is RestFunc-like by FDIFF_1:def 2; :: thesis: verum

end;

hence ( R is RestFunc-like iff for r being Real st r > 0 holds
ex d being Real st
( d > 0 & ( for z being Real st z <> 0 & |.z.| < d holds
|.(R . z).| / |.z.| < r ) ) ) by A1; :: thesis: verum