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)
then A2: r (#) seq is convergent by Th21;

A3: now

assume A4: 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 A5: 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 A4, SEQ_1:13
.= 0 by ABSVALUE:7 ;
hence abs (((r (#) seq) . m) - (r * (lim seq))) < p by A5; :: thesis: verum

end;

now

assume A6: 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 A7: 0 < abs r by COMPLEX1:133;
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 A8: 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
A9: 0 <> abs r by A6, COMPLEX1:133;
0 / (abs r) = 0 ;
then 0 < p / (abs r) by A7, A8, XREAL_1:76;
then consider n1 being Element of NAT such that
A10: 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 A11: abs ((seq . m) - (lim seq)) < p / (abs r) by A10;
A12: abs (((r (#) seq) . m) - (r * (lim seq))) = abs ((r * (seq . m)) - (r * (lim seq))) by SEQ_1:13
.= abs (r * ((seq . m) - (lim seq)))
.= (abs r) * (abs ((seq . m) - (lim seq))) by COMPLEX1:151 ;
(abs r) * (p / (abs r)) = (abs r) * (((abs r) " ) * p) by XCMPLX_0:def 9
.= ((abs r) * ((abs r) " )) * p
.= 1 * p by A9, XCMPLX_0:def 7
.= p ;
hence abs (((r (#) seq) . m) - (r * (lim seq))) < p by A7, A11, A12, XREAL_1:70; :: thesis: verum

end;

hence lim (r (#) seq) = r * (lim seq) by A2, A3, Def7; :: thesis: verum