theorem Th32:
  for l being Linear_Combination of {v} holds Sum(l) = v * l.v
proof
  let l be Linear_Combination of {v};
A1: Carrier(l) c= {v} by Def5;
  now
    per cases by A1,ZFMISC_1:33;
    suppose
      Carrier(l) = {};
      then
A2:   l = ZeroLC(V) by Def4;
      hence Sum(l) = 0.V by Lm3
        .= v * 0.R by VECTSP_2:32
        .= v * l.v by A2,Th18;
    end;
    suppose
      Carrier(l) = {v};
      then consider F such that
A3:   F is one-to-one & rng F = {v} and
A4:   Sum(l) = Sum(l (#) F) by Def7;
      F = <* v *> by A3,FINSEQ_3:97;
      then l (#) F = <* v * l.v *> by Th25;
      hence thesis by A4,RLVECT_1:44;
    end;
  end;
  hence thesis;
end;
