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 Th25:
  for A being Subset of V, v being Element of V st v in A holds (l !A).v = l.v
proof
  let A be Subset of V, v be Element of V such that
A1: v in A;
  dom (l!A) = [#]V by FUNCT_2:92;
  then
A2: (dom (l|A)) \/ (dom ((V \ A) --> 0.F)) = [#]V by FUNCT_4:def 1;
  not v in dom ((V \ A) --> 0.F) by A1,XBOOLE_0:def 5;
  then (l!A).v = (l|A).v by A2,FUNCT_4:def 1
    .= l.v by A1,FUNCT_1:49;
  hence thesis;
end;
