reserve K for Ring,
  V1,W1 for VectSp of K;
reserve F for Field,
  V,W for VectSp of F;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;

theorem Th29:
  T@l is Linear_Combination of T .: (Carrier l)
proof
  Carrier (T@l) c= T .: (Carrier l)
  proof
    let w be object such that
A1: w in Carrier (T@l);
    reconsider w as Element of W by A1;
A2: (T@l).w <> 0.F by A1,VECTSP_6:2;
    now
      assume
A3:   T"{w} misses Carrier l;
      then
A4:   l .: T"{w} c= {0.F} by Th28;
      Sum (l .: T"{w}) = 0.F
      proof
        per cases;
        suppose
          l .: T"{w} = {}F;
          hence thesis by RLVECT_2:8;
        end;
        suppose
A5:       l .: T"{w} <> {}F;
A6:       {0.F} c= l .: T"{w}
          proof
            let y be object;
            assume y in {0.F};
            then
A7:         y = 0.F by TARSKI:def 1;
            ex z being object st z in l .: T"{w} by A5,XBOOLE_0:def 1;
            hence thesis by A4,A7,TARSKI:def 1;
          end;
          l .: T"{w} c= {0.F} by A3,Th28;
          then l .: T"{w} = {0.F} by A6;
          hence thesis by RLVECT_2:9;
        end;
      end;
      hence contradiction by A2,Def5;
    end;
    then consider x being object such that
A8: x in T"{w} and
A9: x in Carrier l by XBOOLE_0:3;
A10: x in dom T by A8,FUNCT_1:def 7;
A11: T.x in {w} by A8,FUNCT_1:def 7;
    reconsider x as Element of V by A8;
    T.x = w by A11,TARSKI:def 1;
    hence thesis by A9,A10,FUNCT_1:def 6;
  end;
  hence thesis by VECTSP_6:def 4;
end;
