reserve k,j,n for Nat,
  r for Real;
reserve x,x1,x2,y for Element of REAL n;
reserve f for real-valued FinSequence;

theorem
  |.r*f.| = |.r.|*|.f.|
proof
  set n = len f;
  reconsider x=f as Element of REAL n by Lm1;
A1: 0 <= (|.r.|)^2 & 0 <= Sum sqr abs x by RVSUM_1:86,XREAL_1:63;
  thus |.r*f.| = sqrt Sum sqr abs(r*x) by Lm2
    .= sqrt Sum sqr (|.r.|*abs x) by Th3
    .= sqrt Sum ((|.r.|)^2 * sqr abs x) by RVSUM_1:58
    .= sqrt ((|.r.|)^2 * Sum sqr abs x) by RVSUM_1:87
    .= sqrt (|.r.|)^2 * sqrt Sum sqr abs x by A1,SQUARE_1:29
    .= |.r.| * sqrt Sum sqr abs x by COMPLEX1:46,SQUARE_1:22
    .= |.r.|*|.f.| by Lm2;
end;
