reserve i,j,m,n,k for Nat,
  x,y for set,
  K for Field,
  a for Element of K;
reserve V for non trivial VectSp of K,
  V1,V2 for VectSp of K,
  f for linear-transformation of V1,V1,
  v,w for Vector of V,
  v1 for Vector of V1,
  L for Scalar of K;
reserve S for 1-sorted,
  F for Function of S,S;

theorem Th20:
  F |^(i+j) = (F |^ j) * (F |^ i)
proof
  set G=GFuncs the carrier of S;
  reconsider Fg=F as Element of G by MONOID_0:73;
  reconsider G as associative unital non empty multMagma;
  reconsider F9=F as Element of G by MONOID_0:73;
  Product((i+j)|->F9) = Product((i|->F9)^(j|->F9)) by FINSEQ_2:123
    .= (Product(i|->F9))*(Product(j|->F9)) by GROUP_4:5
    .= Product(j|->Fg)(*)Product(i|->Fg) by MONOID_0:70
    .= (F|^j)(*)Product(i|->Fg) by Def4
    .= (F|^j)(*)(F|^i) by Def4;
  hence thesis by Def4;
end;
