let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL holds
( f is_convergent_in x0 iff ( ( 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 g being Real st
for g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_convergent_in x0 iff ( ( 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 g being Real st
for g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) )
thus ( f is_convergent_in x0 implies ( ( 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 g being Real st
for g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ) ) :: thesis: ( ( 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 g being Real st
for g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) implies f is_convergent_in x0 )

proof

assume that
A1: f is_convergent_in x0 and
A2: ( ex r1, r2 being Real st
( r1 < x0 & x0 < r2 & ( for g1, g2 being Real holds
( not r1 < g1 or not g1 < x0 or not g1 in dom f or not g2 < r2 or not x0 < g2 or not g2 in dom f ) ) ) or for g being Real ex g1 being Real st
( 0 < g1 & ( for g2 being Real st 0 < g2 holds
ex r1 being Real st
( 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f & |.((f . r1) - g).| >= g1 ) ) ) ) ; :: thesis: contradiction
consider g being Real such that
A3: for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom f) \ {x0} holds
( f /* seq is convergent & lim (f /* seq) = g ) by A1;
consider g1 being Real such that
A4: 0 < g1 and
A5: for g2 being Real st 0 < g2 holds
ex r1 being Real st
( 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f & |.((f . r1) - g).| >= g1 ) by A1, A2;
defpred S₁[ Element of NAT , Real] means ( 0 < |.(x0 - $2).| & |.(x0 - $2).| < 1 / ($1 + 1) & $2 in dom f & |.((f . $2) - g).| >= g1 );
A6: for n being Element of NAT ex r1 being Element of REAL st S₁[n,r1]

proof

let n be Element of NAT ; :: thesis: ex r1 being Element of REAL st S₁[n,r1]
consider r1 being Real such that
A7: S₁[n,r1] by A5, XREAL_1:139;
reconsider r1 = r1 as Element of REAL by XREAL_0:def 1;
take r1 ; :: thesis: S₁[n,r1]
thus S₁[n,r1] by A7; :: thesis: verum

end;

consider s being Real_Sequence such that
A8: for n being Element of NAT holds S₁[n,s . n] from FUNCT_2:sch 3(A6);
A9: rng s c= (dom f) \ {x0} by A8, Th2;
A10: lim s = x0 by A8, Th2;
A11: s is convergent by A8, Th2;
then A12: lim (f /* s) = g by A3, A10, A9;
f /* s is convergent by A3, A11, A10, A9;
then consider n being Nat such that
A13: for k being Nat st n <= k holds
|.(((f /* s) . k) - g).| < g1 by A4, A12, SEQ_2:def 7;
A14: |.(((f /* s) . n) - g).| < g1 by A13;
A15: n in NAT by ORDINAL1:def 12;
rng s c= dom f by A8, Th2;
then |.((f . (s . n)) - g).| < g1 by A14, FUNCT_2:108, A15;
hence contradiction by A8, A15; :: thesis: verum

end;

assume A16: 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 ) ; :: thesis: ( for g being Real ex g1 being Real st
( 0 < g1 & ( for g2 being Real holds
( not 0 < g2 or ex r1 being Real st
( 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f & not |.((f . r1) - g).| < g1 ) ) ) ) or f is_convergent_in x0 )
given g being Real such that A17: for g1 being Real st 0 < g1 holds
ex g2 being Real st
( 0 < g2 & ( for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 ) ) ; :: thesis: f is_convergent_in x0

now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} holds
( f /* s is convergent & lim (f /* s) = g )

let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies ( f /* s is convergent & lim (f /* s) = g ) )
assume that
A18: s is convergent and
A19: lim s = x0 and
A20: rng s c= (dom f) \ {x0} ; :: thesis: ( f /* s is convergent & lim (f /* s) = g )

A21: now :: thesis: for g1 being Real st 0 < g1 holds
ex n being Nat st
for k being Nat st n <= k holds
|.(((f /* s) . k) - g).| < g1

let g1 be Real; :: thesis: ( 0 < g1 implies ex n being Nat st
for k being Nat st n <= k holds
|.(((f /* s) . k) - g).| < g1 )
assume A22: 0 < g1 ; :: thesis: ex n being Nat st
for k being Nat st n <= k holds
|.(((f /* s) . k) - g).| < g1
consider g2 being Real such that
A23: 0 < g2 and
A24: for r1 being Real st 0 < |.(x0 - r1).| & |.(x0 - r1).| < g2 & r1 in dom f holds
|.((f . r1) - g).| < g1 by A17, A22;
consider n being Element of NAT such that
A25: for k being Element of NAT st n <= k holds
( 0 < |.(x0 - (s . k)).| & |.(x0 - (s . k)).| < g2 & s . k in dom f ) by A18, A19, A20, A23, Th3;
reconsider n = n as Nat ;
take n = n; :: thesis: for k being Nat st n <= k holds
|.(((f /* s) . k) - g).| < g1
let k be Nat; :: thesis: ( n <= k implies |.(((f /* s) . k) - g).| < g1 )
A26: k in NAT by ORDINAL1:def 12;
assume A27: n <= k ; :: thesis: |.(((f /* s) . k) - g).| < g1
then A28: |.(x0 - (s . k)).| < g2 by A25, A26;
A29: s . k in dom f by A25, A27, A26;
0 < |.(x0 - (s . k)).| by A25, A27, A26;
then |.((f . (s . k)) - g).| < g1 by A24, A28, A29;
hence |.(((f /* s) . k) - g).| < g1 by A20, FUNCT_2:108, XBOOLE_1:1, A26; :: thesis: verum

end;

hence f /* s is convergent by SEQ_2:def 6; :: thesis: lim (f /* s) = g
hence lim (f /* s) = g by A21, SEQ_2:def 7; :: thesis: verum

end;

hence f is_convergent_in x0 by A16; :: thesis: verum