 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, h being Form of V,W
  holds f+g+h = f+(g+h)
  proof
    let V, W be non empty ModuleStr over INT.Ring, f, g, h be Form of V,W;
  now
    let v be Vector of V, w be Vector of W;
    thus (f+g+h).(v,w) = (f+g).(v,w) + h.(v,w) by BLDef2
      .= f.(v,w) + g.(v,w)+ h.(v,w) by BLDef2
      .= f.(v,w) + (g.(v,w)+ h.(v,w))
      .= f.(v,w) + (g+h).(v,w) by BLDef2
      .= (f+ (g+h)).(v,w) by BLDef2;
    end;
    hence thesis;
  end;
