
theorem Th17:
  for X be non trivial RealNormSpace ex w be VECTOR of X st ||. w .|| = 1
proof
  let X be non trivial RealNormSpace;
  consider v be VECTOR of X such that
A1: v <> 0.X by STRUCT_0:def 18;
  set a= ||. v .||;
  reconsider w=a"*v as VECTOR of X;
  take w;
A2: ||. v .|| <> 0 by A1,NORMSP_0:def 5;
  then
A3: 0 < ||. v .|| by NORMSP_1:4;
A4: |.a".| =|.1*a".| .=|.1/a.| by XCMPLX_0:def 9
    .=1/|.a.| by ABSVALUE:7
    .=1*|.a.|" by XCMPLX_0:def 9
    .=a"by A3,ABSVALUE:def 1;
  thus ||.w.|| =|.a".|*||.v.|| by NORMSP_1:def 1
    .=1 by A2,A4,XCMPLX_0:def 7;
end;
