 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
  for V, W being non empty ModuleStr over INT.Ring, f, g being Form of V,W,
  w being Vector of W
  holds FunctionalSAF(f-g,w) = FunctionalSAF(f,w) - FunctionalSAF(g,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f,g be Form of V,W,
      w be Vector of W;
    now
      let v be Vector of V;
      thus (FunctionalSAF(f-g,w)).v = (f-g).(v,w) by BLTh9
      .= f.(v,w) - g.(v,w) by BLDef7
      .= (FunctionalSAF(f,w)).v - g.(v,w) by BLTh9
      .= (FunctionalSAF(f,w)).v - (FunctionalSAF(g,w)).v by BLTh9
      .= (FunctionalSAF(f,w)).v + (-FunctionalSAF(g,w)).v by HAHNBAN1:def 4
      .= (FunctionalSAF(f,w) -FunctionalSAF(g,w)).v by HAHNBAN1:def 3;
    end;
    hence thesis by FUNCT_2:63;
  end;
