reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem Th38:
  ||.a * x.|| = |.a.| * ||.x.||
proof
A1: 0 <= |.a*a.| by COMPLEX1:46;
  0 <= Re (x.|.x) by Def11;
  then
A2: 0 <= |.(x.|.x).| by Th29;
  ||.a*x.|| = sqrt |.(a*(x.|.(a*x))).| by Def11
    .= sqrt |.(a*(a*' *(x.|.x))).| by Th13
    .= sqrt |.((a*a*')*(x.|.x)).|
    .= sqrt (|.(a*a*').|*|.(x.|.x).|) by COMPLEX1:65
    .= sqrt (|.a*a.| * |.(x.|.x).|) by COMPLEX1:69
    .= sqrt |.a*a.| * sqrt |.(x.|.x).| by A1,A2,SQUARE_1:29
    .= sqrt (|.a.|^2) * sqrt |.(x.|.x).| by COMPLEX1:65
    .= |.a.| * sqrt |.(x.|.x).| by COMPLEX1:46,SQUARE_1:22;
  hence thesis;
end;
