
theorem Th37:
  for V,W be add-associative right_zeroed right_complementable
vector-distributive scalar-distributive scalar-associative scalar-unital
 non empty ModuleStr over F_Complex for 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 add-associative right_zeroed right_complementable
  vector-distributive scalar-distributive scalar-associative scalar-unital
  non empty ModuleStr over F_Complex;
  let 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
BILINEAR:28
    .= 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;
