let c be Real; :: thesis: for seq being Real_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ) holds
( rseq is convergent & lim rseq = abs ((lim seq) - c) )
reconsider cseq = NAT --> c as Real_Sequence ;
let seq be Real_Sequence; :: thesis: ( seq is convergent implies for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ) holds
( rseq is convergent & lim rseq = abs ((lim seq) - c) ) )
assume A1: seq is convergent ; :: thesis: for rseq being Real_Sequence st ( for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ) holds
( rseq is convergent & lim rseq = abs ((lim seq) - c) )
A2: seq - cseq is convergent by A1, SEQ_2:25;
then A3: abs (seq - cseq) is convergent by SEQ_4:26;
let rseq be Real_Sequence; :: thesis: ( ( for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ) implies ( rseq is convergent & lim rseq = abs ((lim seq) - c) ) )
assume A4: for i being Element of NAT holds rseq . i = abs ((seq . i) - c) ; :: thesis: ( rseq is convergent & lim rseq = abs ((lim seq) - c) )

now

let i be Element of NAT ; :: thesis: rseq . i = (abs (seq - cseq)) . i
thus rseq . i = abs ((seq . i) - c) by A4
.= abs ((seq . i) - (cseq . i)) by FUNCOP_1:13
.= abs ((seq . i) + (- (cseq . i)))
.= abs ((seq . i) + ((- cseq) . i)) by SEQ_1:14
.= abs ((seq + (- cseq)) . i) by SEQ_1:11
.= abs ((seq - cseq) . i) by SEQ_1:15
.= (abs (seq - cseq)) . i by SEQ_1:16 ; :: thesis: verum

end;

then A5: for x being set st x in NAT holds
rseq . x = (abs (seq - cseq)) . x ;
lim (abs (seq - cseq)) = abs (lim (seq - cseq)) by A2, SEQ_4:27
.= abs ((lim seq) - (lim cseq)) by A1, SEQ_2:26
.= abs ((lim seq) - (cseq . 0 )) by SEQ_4:41
.= abs ((lim seq) - c) by FUNCOP_1:13 ;
hence ( rseq is convergent & lim rseq = abs ((lim seq) - c) ) by A5, A3, FUNCT_2:18; :: thesis: verum