reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th26:
  f (#) <* v1,v2 *> = <* v1 * f.v1, v2 * f.v2 *>
proof
A1: len(f (#) <* v1,v2 *>) = len<* v1,v2 *> by Def6
    .= 2 by FINSEQ_1:44;
  then
A2: dom(f (#) <* v1,v2 *>) = {1,2} by FINSEQ_1:2,def 3;
  2 in {1,2} by TARSKI:def 2;
  then
A3: (f (#) <* v1,v2 *>).2 = (<* v1,v2 *>/.2) * f.(<* v1,v2 *>/.2) by A2,Def6
    .= v2 * f.(<* v1,v2 *>/.2) by FINSEQ_4:17
    .= v2 * f.v2 by FINSEQ_4:17;
  1 in {1,2} by TARSKI:def 2;
  then (f (#) <* v1,v2 *>).1 = (<* v1,v2 *>/.1) * f.(<* v1,v2 *>/.1) by A2,Def6
    .= v1 * f.(<* v1,v2 *>/.1) by FINSEQ_4:17
    .= v1 * f.v1 by FINSEQ_4:17;
  hence thesis by A1,A3,FINSEQ_1:44;
end;
