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 Th28:
  for X being Subset of V st X misses Carrier l holds l .: X c= { 0.F}
proof
  let X be Subset of V such that
A1: X misses Carrier l;
  set M = l .: X;
  let y be object;
  assume y in M;
  then consider x being object such that
A2: x in dom l and
A3: x in X and
A4: y = l.x by FUNCT_1:def 6;
  reconsider x as Element of V by A2;
  now
    assume l.x <> 0.F;
    then x in Carrier l;
    then x in (Carrier l) /\ X by A3,XBOOLE_0:def 4;
    hence contradiction by A1;
  end;
  hence thesis by A4,TARSKI:def 1;
end;
