theorem
  |.c*z.| = |.c.|*|.z.|
proof
A1: 0 <= |.c.|^2 & 0 <= Sum sqr abs z by RVSUM_1:86,XREAL_1:63;
  thus |.c*z.| = sqrt Sum sqr (|.c.|*abs z) by Th91
    .= sqrt Sum (|.c.|^2 * sqr abs z) by RVSUM_1:58
    .= sqrt (|.c.|^2 * Sum sqr abs z) by RVSUM_1:87
    .= sqrt |.c.|^2 * sqrt Sum sqr abs z by A1,SQUARE_1:29
    .= |.c.|*|.z.| by COMPLEX1:46,SQUARE_1:22;
end;
