let x be Real; :: thesis: for vx being Element of (REAL-NS 1) st vx = <*x*> holds
||.vx.|| = |.x.|
let vx be Element of (REAL-NS 1); :: thesis: ( vx = <*x*> implies ||.vx.|| = |.x.| )
reconsider xx = vx as Element of REAL 1 by REAL_NS1:def 4;
reconsider x2 = x ^2 as Element of REAL by XREAL_0:def 1;
assume vx = <*x*> ; :: thesis: ||.vx.|| = |.x.|
then sqrt (Sum (sqr xx)) = sqrt (Sum <*x2*>) by RVSUM_1:55;
then A1: sqrt (Sum (sqr xx)) = sqrt (x ^2) by RVSUM_1:73;
||.vx.|| = |.xx.| by REAL_NS1:1;
hence ||.vx.|| = |.x.| by A1, COMPLEX1:72; :: thesis: verum