 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 BLTh37:
  for V, W being Z_Module, v, u being Vector of V, w, t being Vector of W,
  f being bilinear-Form of V,W
  holds f.(v-u,w-t) = f.(v,w) - f.(v,t) -(f.(u,w) - f.(u,t))
  proof
    let V, W be Z_Module;
    let v, w be Vector of V, y, z be Vector of W, f be bilinear-Form of V,W;
    set v1 = f.(v,y), v3 = f.(v,z), v4 = f.(w,y), v5 = f.(w,z);
    thus f.(v-w,y-z) = f.(v,y-z) - f.(w,y-z) by BLTh35
    .= v1 - v3 - f.(w,y-z) by BLTh36
    .= v1 - v3 - (v4 - v5) by BLTh36;
  end;
