 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem BLTh24:
  for V, W being non empty ModuleStr over INT.Ring,
  f being Functional of V,
  g being Functional of W, v being Vector of V
  holds FunctionalFAF(FormFunctional(f,g),v) = f.v * g
  proof
    let V, W be non empty ModuleStr over INT.Ring;
    let f be Functional of V,
    h be Functional of W, v be Vector of V;
    set F = FormFunctional(f,h), FF = FunctionalFAF(F,v);
    now
      let y be Vector of W;
      thus FF.y = F.(v,y) by BLTh8
      .= f.v * h.y by BLDef10
      .= (f.v * h).y by HAHNBAN1:def 6;
    end;
    hence thesis by FUNCT_2:63;
  end;
