reserve x,y,y1,y2 for set,
  p for FinSequence,
  i,k,l,n for Nat,
  V for RealLinearSpace,
  u,v,v1,v2,v3,w for VECTOR of V,
  a,b for Real,
  F,G,H1,H2 for FinSequence of V,
  A,B for Subset of V,
  f for Function of the carrier of V, REAL;
reserve K,L,L1,L2,L3 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

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;
