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

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

A6: now :: thesis: ( n1 <= k implies ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p )

assume A7: n1 <= k ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p
take n = 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 A8: n <= m ; :: thesis: |.((seq1 . m) - (lim seq)).| < p
then n1 <= m by A7, XXREAL_0:2;
then |.((seq . m) - (lim seq)).| < p by A5;
hence |.((seq1 . m) - (lim seq)).| < p by A3, A8; :: thesis: verum

end;

now :: thesis: ( k <= n1 implies ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p )

assume A9: k <= n1 ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p
take n = n1; :: 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 A10: n <= m ; :: thesis: |.((seq1 . m) - (lim seq)).| < p
then seq1 . m = seq . m by A3, A9, XXREAL_0:2;
hence |.((seq1 . m) - (lim seq)).| < p by A5, A10; :: thesis: verum

end;

hence ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p by A6; :: thesis: verum

end;

seq1 is convergent by A1, A2, Th19;
hence lim seq = lim seq1 by A4, COMSEQ_2:def 6; :: thesis: verum