 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 BLTh27:
  for V, W being non empty ModuleStr over INT.Ring, v being Vector of V,
  u, w being Vector of W, f being Form of V,W st f is additiveFAF
  holds f.(v,u+w) = f.(v,u) + f.(v,w)
  proof
    let V, W be non empty ModuleStr over INT.Ring;
    let v be Vector of V, y, z be Vector of W, f be Form of V,W;
    set F = FunctionalFAF(f,v);
    assume f is additiveFAF; then
    A1: F is additive;
    thus f.(v,y+z) = F.(y+z) by BLTh8
    .= F.y+F.z by A1
    .= f.(v,y) + F.z by BLTh8
    .= f.(v,y) + f.(v,z) by BLTh8;
  end;
