assume that
A2: lim (f,x0) = g and
A3: 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 & g1 <= |.((f . r1) - g).| ) ) ) ; :: thesis: contradiction
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 & g1 <= |.((f . r1) - g).| ) by A3;
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] 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 A1, A2, A10, A9, Def4;
f /* s is convergent by A1, 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

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
A17: s is convergent and
A18: lim s = x0 and
A19: rng s c= (dom f) \ {x0} ; :: thesis: ( f /* s is convergent & lim (f /* s) = g )
hence f /* s is convergent by SEQ_2:def 6; :: thesis: lim (f /* s) = g
hence lim (f /* s) = g by A20, SEQ_2:def 7; :: thesis: verum