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