reserve p,q,r for FinSequence,
  x,y,y1,y2 for set,
  i,k for Element of NAT,
  GF for add-associative right_zeroed right_complementable Abelian associative
  well-unital distributive non empty doubleLoopStr,
  V for Abelian
  add-associative right_zeroed right_complementable vector-distributive
  scalar-distributive scalar-associative scalar-unital
   non empty ModuleStr over GF,
  u,v,v1,v2,v3,w for Element of V,
  a,b for Element of GF,
  F,G ,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, GF;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;

theorem Th10:
  f (#) <* v *> = <* f.v * v *>
proof
A1: 1 in {1} by TARSKI:def 1;
A2: len(f (#) <* v *>) = len<* v *> by Def5
    .= 1 by FINSEQ_1:40;
  then dom(f (#) <* v *>) = {1} by FINSEQ_1:2,def 3;
  then (f (#) <* v *>).1 = f.(<* v *>/.1) * <* v *>/.1 by A1,Def5
    .= f.(<* v *>/.1) * v by FINSEQ_4:16
    .= f.v * v by FINSEQ_4:16;
  hence thesis by A2,FINSEQ_1:40;
end;
