theorem Th32:
  for l being Linear_Combination of {v} holds Sum(l) = l.v * v
proof
  let l be Linear_Combination of {v};
A1: Carrier(l) c= {v} by Def6;
  now
    per cases by A1,ZFMISC_1:33;
    suppose
      Carrier(l) = {};
      then
A2:   l = ZeroLC(V) by Def5;
      hence Sum(l) = 0.V by Lm2
        .= 0 * v by RLVECT_1:10
        .= l.v * v by A2,Th20;
    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 Def8;
      F = <* v *> by A3,FINSEQ_3:97;
      then l (#) F = <* l.v * v *> by Th26;
      hence thesis by A4,RLVECT_1:44;
    end;
  end;
  hence thesis;
end;
