let c be Real; :: thesis: for seq being Real_Sequence st seq is convergent holds
for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| )
reconsider cc = c as Element of REAL by XREAL_0:def 1;
set cseq = seq_const c;
let seq be Real_Sequence; :: thesis: ( seq is convergent implies for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| ) )
assume A1: seq is convergent ; :: thesis: for rseq being Real_Sequence st ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) holds
( rseq is convergent & lim rseq = |.((lim seq) - c).| )
A2: seq - (seq_const c) is convergent by A1;
then A3: abs (seq - (seq_const c)) is convergent ;
let rseq be Real_Sequence; :: thesis: ( ( for i being Nat holds rseq . i = |.((seq . i) - c).| ) implies ( rseq is convergent & lim rseq = |.((lim seq) - c).| ) )
assume A4: for i being Nat holds rseq . i = |.((seq . i) - c).| ; :: thesis: ( rseq is convergent & lim rseq = |.((lim seq) - c).| )

now :: thesis: for i being Nat holds rseq . i = (abs (seq - (seq_const c))) . i

let i be Nat; :: thesis: rseq . i = (abs (seq - (seq_const c))) . i
thus rseq . i = |.((seq . i) - c).| by A4
.= |.((seq . i) - ((seq_const c) . i)).| by SEQ_1:57
.= |.((seq . i) + (- ((seq_const c) . i))).|
.= |.((seq . i) + ((- (seq_const c)) . i)).| by SEQ_1:10
.= |.((seq + (- (seq_const c))) . i).| by SEQ_1:7
.= |.((seq - (seq_const c)) . i).| by SEQ_1:11
.= (abs (seq - (seq_const c))) . i by SEQ_1:12 ; :: thesis: verum

end;

then A5: for x being object st x in NAT holds
rseq . x = (abs (seq - (seq_const c))) . x ;
lim (abs (seq - (seq_const c))) = |.(lim (seq - (seq_const c))).| by A2, SEQ_4:14
.= |.((lim seq) - (lim (seq_const c))).| by A1, SEQ_2:12
.= |.((lim seq) - ((seq_const c) . 0)).| by SEQ_4:26
.= |.((lim seq) - c).| by SEQ_1:57 ;
hence ( rseq is convergent & lim rseq = |.((lim seq) - c).| ) by A5, A3, FUNCT_2:12; :: thesis: verum