let a, b be Real_Sequence; :: thesis: for c being convergent Complex_Sequence st ( for n being Nat holds
( Re (c . n) = a . n & Im (c . n) = b . n ) ) holds
( a is convergent & b is convergent & lim a = Re (lim c) & lim b = Im (lim c) )
let c be convergent Complex_Sequence; :: thesis: ( ( for n being Nat holds
( Re (c . n) = a . n & Im (c . n) = b . n ) ) implies ( a is convergent & b is convergent & lim a = Re (lim c) & lim b = Im (lim c) ) )
assume A1: for n being Nat holds
( Re (c . n) = a . n & Im (c . n) = b . n ) ; :: thesis: ( a is convergent & b is convergent & lim a = Re (lim c) & lim b = Im (lim c) )
A2: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((b . m) - (Im (lim c))).| < p

proof

let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.((b . m) - (Im (lim c))).| < p )
assume A3: p > 0 ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((b . m) - (Im (lim c))).| < p
consider n being Nat such that
A4: for m being Nat st n <= m holds
|.((c . m) - (lim c)).| < p by A3, COMSEQ_2:def 6;
take n ; :: thesis: for m being Nat st n <= m holds
|.((b . m) - (Im (lim c))).| < p
let m be Nat; :: thesis: ( n <= m implies |.((b . m) - (Im (lim c))).| < p )
assume n <= m ; :: thesis: |.((b . m) - (Im (lim c))).| < p
then A5: |.((c . m) - (lim c)).| < p by A4;
( Im (c . m) = b . m & Im ((c . m) - (lim c)) = (Im (c . m)) - (Im (lim c)) ) by A1, COMPLEX1:19;
then |.((b . m) - (Im (lim c))).| <= |.((c . m) - (lim c)).| by Th13;
hence |.((b . m) - (Im (lim c))).| < p by A5, XXREAL_0:2; :: thesis: verum

end;

A6: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((a . m) - (Re (lim c))).| < p

proof

let p be Real; :: thesis: ( 0 < p implies ex n being Nat st
for m being Nat st n <= m holds
|.((a . m) - (Re (lim c))).| < p )
assume A7: 0 < p ; :: thesis: ex n being Nat st
for m being Nat st n <= m holds
|.((a . m) - (Re (lim c))).| < p
consider n being Nat such that
A8: for m being Nat st n <= m holds
|.((c . m) - (lim c)).| < p by A7, COMSEQ_2:def 6;
take n ; :: thesis: for m being Nat st n <= m holds
|.((a . m) - (Re (lim c))).| < p
let m be Nat; :: thesis: ( n <= m implies |.((a . m) - (Re (lim c))).| < p )
Re (c . m) = a . m by A1;
then Re ((c . m) - (lim c)) = (a . m) - (Re (lim c)) by COMPLEX1:19;
then A9: |.((a . m) - (Re (lim c))).| <= |.((c . m) - (lim c)).| by Th13;
assume n <= m ; :: thesis: |.((a . m) - (Re (lim c))).| < p
then |.((c . m) - (lim c)).| < p by A8;
hence |.((a . m) - (Re (lim c))).| < p by A9, XXREAL_0:2; :: thesis: verum

end;

( a is convergent & b is convergent ) by A1, Th38;
hence ( a is convergent & b is convergent & lim a = Re (lim c) & lim b = Im (lim c) ) by A6, A2, SEQ_2:def 7; :: thesis: verum