let z be Complex; :: thesis: for CNS being ComplexNormSpace
for S being sequence of CNS st S is convergent holds
lim (z * S) = z * (lim S)
let CNS be ComplexNormSpace; :: thesis: for S being sequence of CNS st S is convergent holds
lim (z * S) = z * (lim S)
let S be sequence of CNS; :: thesis: ( S is convergent implies lim (z * S) = z * (lim S) )
assume A1: S is convergent ; :: thesis: lim (z * S) = z * (lim S)
then A2: z * S is convergent by Th118;
set g = lim S;
set h = z * (lim S);

A3: now

assume A4: z = 0 ; :: thesis: for r being Real st 0 < r holds
ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
let r be Real; :: thesis: ( 0 < r implies ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r )
assume 0 < r ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
then consider m1 being Element of NAT such that
A5: for n being Element of NAT st m1 <= n holds
||.((S . n) - (lim S)).|| < r by A1, Def18;
take k = m1; :: thesis: for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
let n be Element of NAT ; :: thesis: ( k <= n implies ||.(((z * S) . n) - (z * (lim S))).|| < r )
assume k <= n ; :: thesis: ||.(((z * S) . n) - (z * (lim S))).|| < r
then A6: ||.((S . n) - (lim S)).|| < r by A5;
||.((z * (S . n)) - (z * (lim S))).|| = ||.((0c * (S . n)) - H₁(CNS)).|| by A4, Th2
.= ||.(H₁(CNS) - H₁(CNS)).|| by Th2
.= ||.H₁(CNS).|| by RLVECT_1:26
.= 0 by Def11 ;
then ||.((z * (S . n)) - (z * (lim S))).|| < r by A6, Th106;
hence ||.(((z * S) . n) - (z * (lim S))).|| < r by Def15; :: thesis: verum

end;

now

assume A7: z <> 0c ; :: thesis: for r being Real st 0 < r holds
ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
then A8: 0 < |.z.| by COMPLEX1:133;
let r be Real; :: thesis: ( 0 < r implies ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r )
assume A9: 0 < r ; :: thesis: ex k being Element of NAT st
for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
A10: 0 <> |.z.| by A7, COMPLEX1:133;
0 < r / |.z.| by A8, A9;
then consider m1 being Element of NAT such that
A11: for n being Element of NAT st m1 <= n holds
||.((S . n) - (lim S)).|| < r / |.z.| by A1, Def18;
take k = m1; :: thesis: for n being Element of NAT st k <= n holds
||.(((z * S) . n) - (z * (lim S))).|| < r
let n be Element of NAT ; :: thesis: ( k <= n implies ||.(((z * S) . n) - (z * (lim S))).|| < r )
assume k <= n ; :: thesis: ||.(((z * S) . n) - (z * (lim S))).|| < r
then A12: ||.((S . n) - (lim S)).|| < r / |.z.| by A11;
A13: ||.((z * (S . n)) - (z * (lim S))).|| = ||.(z * ((S . n) - (lim S))).|| by Th10
.= |.z.| * ||.((S . n) - (lim S)).|| by Def11 ;
|.z.| * (r / |.z.|) = |.z.| * ((|.z.| " ) * r) by XCMPLX_0:def 9
.= (|.z.| * (|.z.| " )) * r
.= 1 * r by A10, XCMPLX_0:def 7
.= r ;
then ||.((z * (S . n)) - (z * (lim S))).|| < r by A8, A12, A13, XREAL_1:70;
hence ||.(((z * S) . n) - (z * (lim S))).|| < r by Def15; :: thesis: verum

end;

hence lim (z * S) = z * (lim S) by A2, A3, Def18; :: thesis: verum