 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 BLTh8:
  for V, W being non empty ModuleStr over INT.Ring, f being Form of V,W,
  v being Vector of V
  holds dom (FunctionalFAF(f,v)) = the carrier of W &
  rng (FunctionalFAF(f,v)) c= the carrier of INT.Ring &
  for w be Vector of W holds (FunctionalFAF(f,v)).w = f.(v,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring;
    let f be Form of V,W, v be Vector of V;
    set F = FunctionalFAF(f,v);
    dom f = [:the carrier of V,the carrier of W:] by FUNCT_2:def 1;
    then
    A1: ex g being Function st (curry f).v = g & dom g = the carrier of W &
    rng g c= rng f & for y being object st y in the carrier of W holds
    g.y = f.(v,y) by FUNCT_5:29;
    hence dom F = the carrier of W & rng F c= the carrier of INT.Ring;
    let y be Vector of W;
    thus thesis by A1;
  end;
