let seq, seq1 be Complex_Sequence; :: thesis: ( seq is convergent & ex k being Element of NAT st seq = seq1 ^\ k implies seq1 is convergent )
assume that
A1: seq is convergent and
A2: ex k being Element of NAT st seq = seq1 ^\ k ; :: thesis: seq1 is convergent
consider k being Element of NAT such that
A3: seq = seq1 ^\ k by A2;
consider g1 being Element of COMPLEX such that
A4: for p being Real st 0 < p holds
ex n being Element of NAT st
for m being Element of NAT st n <= m holds
|.((seq . m) - g1).| < p by A1, COMSEQ_2:def 5;
take g = g1; :: according to COMSEQ_2:def 5 :: thesis: for b₁ being Element of REAL holds
( b₁ <= 0 or ex b₂ being Element of NAT st
for b₃ being Element of NAT holds
( not b₂ <= b₃ or not b₁ <= |.K65((seq1 . b₃),g).| ) )
let p be Real; :: thesis: ( p <= 0 or ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= |.K65((seq1 . b₂),g).| ) )
assume 0 < p ; :: thesis: ex b₁ being Element of NAT st
for b₂ being Element of NAT holds
( not b₁ <= b₂ or not p <= |.K65((seq1 . b₂),g).| )
then consider n1 being Element of NAT such that
A5: for m being Element of NAT st n1 <= m holds
|.((seq . m) - g1).| < p by A4;
take n = n1 + k; :: thesis: for b₁ being Element of NAT holds
( not n <= b₁ or not p <= |.K65((seq1 . b₁),g).| )
let m be Element of NAT ; :: thesis: ( not n <= m or not p <= |.K65((seq1 . m),g).| )
assume A6: n <= m ; :: thesis: not p <= |.K65((seq1 . m),g).|
then consider l being Nat such that
A7: m = (n1 + k) + l by NAT_1:10;
reconsider l = l as Element of NAT by ORDINAL1:def 12;
m - k = ((n1 + l) + k) + (- k) by A7;
then consider m1 being Element of NAT such that
A8: m1 = m - k ;

now :: thesis: n1 <= m1

assume not n1 <= m1 ; :: thesis: contradiction
then m1 + k < n1 + k by XREAL_1:6;
hence contradiction by A6, A8; :: thesis: verum

end;

then A9: |.((seq . m1) - g1).| < p by A5;
m1 + k = m by A8;
hence |.((seq1 . m) - g).| < p by A3, A9, NAT_1:def 3; :: thesis: verum