let c be Complex; :: thesis: for seq being Complex_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| )
let seq be Complex_Sequence; :: thesis: ( seq is convergent implies for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| ) )
assume A1: seq is convergent ; :: thesis: for rseq being Real_Sequence st ( for m being Element of NAT holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| )
let rseq be Real_Sequence; :: thesis: ( ( for m being Element of NAT holds rseq . m = |.((seq . m) - c).| * |.((seq . m) - c).| ) implies ( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| ) )
assume A2: for i being Element of NAT holds rseq . i = |.((seq . i) - c).| * |.((seq . i) - c).| ; :: thesis: ( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| )
reconsider cseq = NAT --> c as Complex_Sequence ;
A3: for n being Element of NAT holds cseq . n = c by FUNCOP_1:13;

now

let i be Element of NAT ; :: thesis: rseq . i = (|.(seq - cseq).| (#) |.(seq - cseq).|) . i
thus rseq . i = |.((seq . i) - c).| * |.((seq . i) - c).| by A2
.= |.((seq . i) - (cseq . i)).| * |.((seq . i) - c).| by FUNCOP_1:13
.= |.((seq . i) - (cseq . i)).| * |.((seq . i) - (cseq . i)).| by FUNCOP_1:13
.= |.((seq . i) + ((- cseq) . i)).| * |.((seq . i) + (- (cseq . i))).| by VALUED_1:8
.= |.((seq . i) + ((- cseq) . i)).| * |.((seq . i) + ((- cseq) . i)).| by VALUED_1:8
.= |.((seq + (- cseq)) . i).| * |.((seq . i) + ((- cseq) . i)).| by VALUED_1:1
.= |.((seq - cseq) . i).| * |.((seq + (- cseq)) . i).| by VALUED_1:1
.= (|.(seq - cseq).| . i) * |.((seq - cseq) . i).| by VALUED_1:18
.= (|.(seq - cseq).| . i) * (|.(seq - cseq).| . i) by VALUED_1:18
.= (|.(seq - cseq).| (#) |.(seq - cseq).|) . i by SEQ_1:12 ; :: thesis: verum

end;

then A4: for x being set st x in NAT holds
rseq . x = (|.(seq - cseq).| (#) |.(seq - cseq).|) . x ;
A5: cseq is convergent by A3, COMSEQ_2:9;
then reconsider seq1 = seq - cseq as convergent Complex_Sequence by A1, COMSEQ_2:25;
A6: |.seq1.| is convergent ;
A7: |.seq1.| (#) |.seq1.| is convergent by SEQ_2:28;
lim (|.(seq - cseq).| (#) |.(seq - cseq).|) = (lim |.(seq - cseq).|) * (lim |.(seq - cseq).|) by A6, SEQ_2:29
.= |.((lim seq) - (lim cseq)).| * (lim |.(seq - cseq).|) by A1, A5, COMSEQ_2:27
.= |.((lim seq) - (lim cseq)).| * |.((lim seq) - (lim cseq)).| by A1, A5, COMSEQ_2:27
.= |.((lim seq) - c).| * |.((lim seq) - (lim cseq)).| by A3, COMSEQ_2:10
.= |.((lim seq) - c).| * |.((lim seq) - c).| by A3, COMSEQ_2:10 ;
hence ( rseq is convergent & lim rseq = |.((lim seq) - c).| * |.((lim seq) - c).| ) by A4, A7, FUNCT_2:18; :: thesis: verum