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 Th26:
  for A being Subset of V, v being Element of V st not v in A
  holds (l!A).v = 0.F
proof
  let A be Subset of V, v be Element of V;
  assume not v in A;
  then
A1: v in V \ A by XBOOLE_0:def 5;
A2: dom (l!A) = [#]V by FUNCT_2:92;
  dom ((V \ A) --> 0.F) = V \ A & dom (l!A) = (dom (l|A)) \/ (dom ((V \ A)
  --> 0.F)) by FUNCT_4:def 1;
  then (l!A).v = ((V \ A) --> 0.F).v by A2,A1,FUNCT_4:def 1
    .= 0.F by A1,FUNCOP_1:7;
  hence thesis;
end;
