theorem Th33:
  v1 <> v2 implies for l being Linear_Combination of {v1,v2} holds
  Sum (l) = v1 * l.v1 + v2 * l.v2
proof
  assume
A1: v1 <> v2;
  let l be Linear_Combination of {v1,v2};
A2: Carrier(l) c= {v1,v2} by Def5;
  now
    per cases by A2,ZFMISC_1:36;
    suppose
      Carrier(l) = {};
      then
A3:   l = ZeroLC(V) by Def4;
      hence Sum(l) = 0.V by Lm3
        .= 0.V + 0.V by RLVECT_1:def 4
        .= v1 * 0.R + 0.V by VECTSP_2:32
        .= v1 * 0.R + v2 * 0.R by VECTSP_2:32
        .= v1 * l.v1 + v2 * 0.R by A3,Th18
        .= v1 * l.v1 + v2 * l.v2 by A3,Th18;
    end;
    suppose
A4:   Carrier(l) = {v1};
      then reconsider L = l as Linear_Combination of {v1} by Def5;
A5:   not v2 in Carrier(l) by A1,A4,TARSKI:def 1;
      thus Sum(l) = Sum(L) .= v1 * l.v1 by Th32
        .= v1 * l.v1 + 0.V by RLVECT_1:def 4
        .= v1 * l.v1 + v2 * 0.R by VECTSP_2:32
        .= v1 * l.v1 + v2 * l.v2 by A5;
    end;
    suppose
A6:   Carrier(l) = {v2};
      then reconsider L = l as Linear_Combination of {v2} by Def5;
A7:   not v1 in Carrier(l) by A1,A6,TARSKI:def 1;
      thus Sum(l) = Sum(L) .= v2 * l.v2 by Th32
        .= 0.V + v2 * l.v2 by RLVECT_1:def 4
        .= v1 * 0.R + v2 * l.v2 by VECTSP_2:32
        .= v1 * l.v1 + v2 * l.v2 by A7;
    end;
    suppose
      Carrier(l) = {v1,v2};
      then consider F such that
A8:   F is one-to-one & rng F = {v1,v2} and
A9:   Sum(l) = Sum(l (#) F) by Def7;
      F = <* v1,v2 *> or F = <* v2,v1 *> by A1,A8,FINSEQ_3:99;
      then l (#) F = <* v1 * l.v1, v2 * l.v2 *> or l (#) F = <* v2 * l.v2, v1
      * l.v1 *> by Th26;
      hence thesis by A9,RLVECT_1:45;
    end;
  end;
  hence thesis;
end;
