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 Th35:
  for X being Subset of V st X misses Carrier l & X <> {} holds l .: X = {0.F}
proof
  let X be Subset of V such that
A1: X misses Carrier l and
A2: X <> {};
  dom l = [#]V by FUNCT_2:92;
  then
A3: l .: X <> {} by A2,RELAT_1:119;
  l .: X c= {0.F} by A1,Th28;
  hence thesis by A3,ZFMISC_1:33;
end;
