
theorem Th2:
  for x be Real, vx be Element of REAL-NS 1
  st vx = <* x *> holds ||.vx.|| = |.x.|
proof
let x be Real;
let vx be Element of REAL-NS 1;
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 *>; 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 thesis by A1,COMPLEX1:72;
end;
