let p be Real; :: thesis: ( p >= 1 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
( lp is reflexive & lp is discerning & lp is RealNormSpace-like ) )
assume A1: p >= 1 ; :: 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
( lp is reflexive & lp is discerning & lp is RealNormSpace-like )
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 ( lp is reflexive & lp is discerning & lp is RealNormSpace-like ) )
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: ( lp is reflexive & lp is discerning & lp is RealNormSpace-like )
hence ||.(0. lp).|| = 0 by A1, Th13; :: according to NORMSP_0:def 6 :: thesis: ( lp is discerning & lp is RealNormSpace-like )
for x, y being Point of lp
for a being Real holds
( ( ||.x.|| = 0 implies x = 0. lp ) & ( x = 0. lp implies ||.x.|| = 0 ) & 0 <= ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| & ||.(a * x).|| = |.a.| * ||.x.|| ) by A1, Th13, A2;
hence ( lp is discerning & lp is RealNormSpace-like ) by NORMSP_1:def 1; :: thesis: verum