
theorem
  for K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr for V,W be
VectSp of K for v,u be Vector of V, w,t be Vector of W, a,b be Element of K for
f be bilinear-Form of V,W holds f.(v-a*u,w-b*t) = f.(v,w) - b*f.(v,t) - (a*f.(u
  ,w) - a*(b*f.(u,t)))
proof
  let K be add-associative right_zeroed right_complementable Abelian
  associative well-unital distributive non empty doubleLoopStr, V,W be VectSp
  of K, v,w be Vector of V, y,z be Vector of W, a,b be Element of K, 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-a*w,y-b*z) = v1 -f.(v,b*z) - (f.(a*w,y) -f.(a*w,b*z)) by Th37
    .= v1 -b*v3 - (f.(a*w,y) -f.(a*w,b*z)) by Th32
    .= v1 - b*v3 - (a*v4 - f.(a*w,b*z)) by Th31
    .= v1 - b*v3 - (a*v4 - a*f.(w,b*z)) by Th31
    .= v1 - b*v3 - (a*v4 - a*(b*v5)) by Th32;
end;
