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 Th36:
  for w being Element of W st w in Carrier (T@l) holds T"{w} meets Carrier l
proof
  let w be Element of W;
  assume w in Carrier (T@l);
  then
A1: (T@l).w <> 0.F by VECTSP_6:2;
  assume
A2: T"{w} misses Carrier l;
  per cases;
  suppose
    T"{w} = {};
    then Sum (l .: T"{w}) = Sum ({}F)
      .= 0.F by RLVECT_2:8;
    hence thesis by A1,Def5;
  end;
  suppose
    T"{w} <> {};
    then l .: T"{w} = {0.F} by A2,Th35;
    then Sum (l .: T"{w}) = 0.F by RLVECT_2:9;
    hence thesis by A1,Def5;
  end;
end;
