let r be real number ; :: thesis: for seq being Real_Sequence st seq is convergent holds
lim (r (#) seq) = r * (lim seq)
let seq be Real_Sequence; :: thesis: ( seq is convergent implies lim (r (#) seq) = r * (lim seq) )
assume A1: seq is convergent ; :: thesis: lim (r (#) seq) = r * (lim seq)

A2: now

assume A3: r = 0 ; :: thesis: for p being real number st 0 < p holds
ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
let p be real number ; :: thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p )
assume A4: 0 < p ; :: thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
take k = 0 ; :: thesis: for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
let m be Element of NAT ; :: thesis: ( k <= m implies abs (((r (#) seq) . m) - (r * (lim seq))) < p )
assume k <= m ; :: thesis: abs (((r (#) seq) . m) - (r * (lim seq))) < p
abs (((r (#) seq) . m) - (r * (lim seq))) = abs ((0 * (seq . m)) - 0) by A3, SEQ_1:9
.= 0 by ABSVALUE:2 ;
hence abs (((r (#) seq) . m) - (r * (lim seq))) < p by A4; :: thesis: verum

end;

now

assume A5: r <> 0 ; :: thesis: for p being real number st 0 < p holds
ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
then A6: 0 < abs r by COMPLEX1:47;
let p be real number ; :: thesis: ( 0 < p implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p )
assume A7: 0 < p ; :: thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
A8: 0 <> abs r by A5, COMPLEX1:47;
0 / (abs r) = 0 ;
then 0 < p / (abs r) by A6, A7, XREAL_1:74;
then consider n1 being Element of NAT such that
A9: for m being Element of NAT st n1 <= m holds
abs ((seq . m) - (lim seq)) < p / (abs r) by A1, Def7;
take k = n1; :: thesis: for m being Element of NAT st k <= m holds
abs (((r (#) seq) . m) - (r * (lim seq))) < p
let m be Element of NAT ; :: thesis: ( k <= m implies abs (((r (#) seq) . m) - (r * (lim seq))) < p )
assume k <= m ; :: thesis: abs (((r (#) seq) . m) - (r * (lim seq))) < p
then A10: abs ((seq . m) - (lim seq)) < p / (abs r) by A9;
A11: abs (((r (#) seq) . m) - (r * (lim seq))) = abs ((r * (seq . m)) - (r * (lim seq))) by SEQ_1:9
.= abs (r * ((seq . m) - (lim seq)))
.= (abs r) * (abs ((seq . m) - (lim seq))) by COMPLEX1:65 ;
(abs r) * (p / (abs r)) = (abs r) * (((abs r) ") * p) by XCMPLX_0:def 9
.= ((abs r) * ((abs r) ")) * p
.= 1 * p by A8, XCMPLX_0:def 7
.= p ;
hence abs (((r (#) seq) . m) - (r * (lim seq))) < p by A6, A10, A11, XREAL_1:68; :: thesis: verum

end;

hence lim (r (#) seq) = r * (lim seq) by A1, A2, Def7; :: thesis: verum