
theorem Th38:
  for V,W be VectSp of F_Complex, v,u be Vector of V, w,t be
  Vector of W for a,b be Element of F_Complex for f be sesquilinear-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 V,W be VectSp of F_Complex, v1,w1 be Vector of V, w,w2 be Vector of W, r
  ,s be Element of F_Complex, f be sesquilinear-Form of V,W;
  set v3 = f.(v1,w), v4 = f.(v1,w2), v5 = f.(w1,w), v6 = f.(w1,w2);
  thus f.(v1-r*w1,w-s*w2) = v3 -f.(v1,s*w2) - (f.(r*w1,w) -f.(r*w1,s*w2)) by
Th36
    .= v3 -s*'*v4 - (f.(r*w1,w) -f.(r*w1,s*w2)) by Th27
    .= v3 - s*'*v4 - (r*v5 - f.(r*w1,s*w2)) by BILINEAR:31
    .= v3 - s*'*v4 - (r*v5 - r*f.(w1,s*w2)) by BILINEAR:31
    .= v3 - s*'*v4 - (r*v5 - r*(s*'*v6)) by Th27;
end;
