
theorem Th17:
  for X be non trivial ComplexNormSpace ex w be VECTOR of X st ||. w .|| = 1
proof
  let X be non trivial ComplexNormSpace;
  consider v be VECTOR of X such that
A1: v <> 0.X by STRUCT_0:def 18;
  reconsider a= ||. v .||+0*<i> as Element of COMPLEX by XCMPLX_0:def 2;
  take w=a"*v;
A2: ||. v .|| <> 0 by A1,NORMSP_0:def 5;
  then
A3: 0 < ||. v .|| by CLVECT_1:105;
A4: |.a".| = |.(1r*a").| by COMPLEX1:def 4
    .= |.1r/a.| by XCMPLX_0:def 9
    .= 1/|.a.| by COMPLEX1:48,67
    .= 1*|.a.|" by XCMPLX_0:def 9
    .= ||.v.||"by A3,ABSVALUE:def 1;
  thus ||.w.|| =|.a".|*||.v.|| by CLVECT_1:def 13
    .=1 by A2,A4,XCMPLX_0:def 7;
end;
