let p be Real; :: thesis: ( 1 <= p implies for lp being non empty NORMSTR st lp = NORMSTR(# (the_set_of_RealSequences_l^ p),(Zero_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Add_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Mult_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(l_norm^ p) #) holds
for x being Point of lp
for a being Real holds Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p)) )
assume A1: 1 <= p ; :: thesis: for lp being non empty NORMSTR st lp = NORMSTR(# (the_set_of_RealSequences_l^ p),(Zero_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Add_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Mult_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(l_norm^ p) #) holds
for x being Point of lp
for a being Real holds Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p))
let lp be non empty NORMSTR ; :: thesis: ( lp = NORMSTR(# (the_set_of_RealSequences_l^ p),(Zero_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Add_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Mult_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(l_norm^ p) #) implies for x being Point of lp
for a being Real holds Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p)) )
assume A2: lp = NORMSTR(# (the_set_of_RealSequences_l^ p),(Zero_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Add_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(Mult_ ((the_set_of_RealSequences_l^ p),Linear_Space_of_RealSequences)),(l_norm^ p) #) ; :: thesis: for x being Point of lp
for a being Real holds Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p))
let x be Point of lp; :: thesis: for a being Real holds Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p))
A3: (seq_id x) rto_power p is summable by A1, A2, Th10;
let a be Real; :: thesis: Sum ((seq_id (a * x)) rto_power p) = ((abs a) to_power p) * (Sum ((seq_id x) rto_power p))
thus Sum ((seq_id (a * x)) rto_power p) = Sum (((abs a) to_power p) (#) ((seq_id x) rto_power p)) by A1, A2, Lm6
.= ((abs a) to_power p) * (Sum ((seq_id x) rto_power p)) by A3, SERIES_1:10 ; :: thesis: verum