let seq, seq1 be Complex_Sequence; :: thesis: ( seq is convergent & ex k being Nat st seq = seq1 ^\ k implies lim seq1 = lim seq )
assume that
A1: seq is convergent and
A2: ex k being Nat st seq = seq1 ^\ k ; :: thesis: lim seq1 = lim seq
consider k being Nat such that
A3: seq = seq1 ^\ k by A2;

A4: now :: thesis: for p being Real st 0 < p holds
ex n being set st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p

let p be Real; :: thesis: ( 0 < p implies ex n being set st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p )
assume 0 < p ; :: thesis: ex n being set st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p
then consider n1 being Nat such that
A5: for m being Nat st n1 <= m holds
|.((seq . m) - (lim seq)).| < p by A1, COMSEQ_2:def 6;
take n = n1 + k; :: thesis: for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p
let m be Nat; :: thesis: ( n <= m implies |.((seq1 . m) - (lim seq)).| < p )
assume A6: n <= m ; :: thesis: |.((seq1 . m) - (lim seq)).| < p
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 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) - (lim seq)).| < p by A5;
m1 + k = m by A8;
hence |.((seq1 . m) - (lim seq)).| < p by A3, A9, NAT_1:def 3; :: thesis: verum

end;

seq1 is convergent by A1, A2, Th22;
hence lim seq1 = lim seq by A4, COMSEQ_2:def 6; :: thesis: verum