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 Th8:
  i in dom F & v = F.i implies (f (#) F).i = f.v * v
proof
  assume that
A1: i in dom F and
A2: v = F.i;
A3: F/.i = F.i by A1,PARTFUN1:def 6;
  len(f (#) F) = len F by Def5;
  then i in dom(f (#) F) by A1,FINSEQ_3:29;
  hence thesis by A2,A3,Def5;
end;
