let x be Real; :: thesis: for vx being Element of (REAL-NS 2) st vx = <*0,x*> holds
||.vx.|| = |.x.|
let vx be Element of (REAL-NS 2); :: thesis: ( vx = <*0,x*> implies ||.vx.|| = |.x.| )
( x in REAL & 0 in REAL ) by XREAL_0:def 1;
then reconsider xx = <*0,x*> as Element of REAL 2 by FINSEQ_2:101;
reconsider xx1 = xx as Point of (TOP-REAL 2) by EUCLID:22;
assume vx = <*0,x*> ; :: thesis: ||.vx.|| = |.x.|
then A1: ||.vx.|| = |.xx.| by REAL_NS1:1;
xx1 `2 = x by FINSEQ_1:44;
then A2: (sqr xx) . 2 = x ^2 by VALUED_1:11;
xx1 `1 = 0 by FINSEQ_1:44;
then ( len (sqr xx) = 2 & (sqr xx) . 1 = 0 ^2 ) by CARD_1:def 7, VALUED_1:11;
then sqr xx = <*(0 ^2),(x ^2)*> by A2, FINSEQ_1:44;
then sqrt (Sum (sqr xx)) = sqrt ((0 ^2) + (x ^2)) by RVSUM_1:77
.= sqrt ((0 * 0) + (x ^2)) by SQUARE_1:def 1
.= sqrt (x ^2) ;
hence ||.vx.|| = |.x.| by A1, COMPLEX1:72; :: thesis: verum