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 Th25:
  f (#) <* v *> = <* v * f.v *>
proof
A1: 1 in {1} by TARSKI:def 1;
A2: len(f (#) <* v *>) = len<* v *> by Def6
    .= 1 by FINSEQ_1:40;
  then dom(f (#) <* v *>) = {1} by FINSEQ_1:2,def 3;
  then (f (#) <* v *>).1 = (<* v *>/.1) * f.(<* v *>/.1) by A1,Def6
    .= v * f.(<* v *>/.1) by FINSEQ_4:16
    .= v * f.v by FINSEQ_4:16;
  hence thesis by A2,FINSEQ_1:40;
end;
