Sentence examples for the assertion holds true from inspiring English sources

On Γ F the assertion holds true due to (14).

Since the assertion holds true for (A_{1,k}) and (A_{2,l}) in place of (A_1) and (A_2), it follows from the latter estimate that begin{aligned} A_1circ A_2 = sum _{k,l=1}^4i^{k+l}A_{1,k}circ A_{2,l}in mathbb U^{p,q} omega,Lambda ) end{aligned}is uniquely defined and that (2.7) holds for (A_min mathbb U^{p_m,q_m} omega _m,Lambda )), (m=1,2).

If the last assertion holds true, then there exists a (widetilde {K}inmathscr{L}(H_{1};H_{2})) such that (F=Gwidetilde{K}) and (Vert widetilde{K}Vert _{mathscr {L}(H_{1};H_{2}leqslantnt C). (exists Kinmathscr{L}(H_{1};H_{2})) such that (F=GK).

A similar assertion holds true if we switch I j and I j in the above inequalities.

In fact, we prove that an even stronger assertion holds true.

Theorem 1.2 With the above, the following assertions hold true: (1) The arithmetic, geometric and harmonic means A, G and H are stable.

Proposition 3.4 Let be a Banach algebra with a unit e, P be a cone in and ⪯ be the semi-order generated by the cone P. The following assertions hold true: (i) For any x, y ∈ A, a ∈ P with x ⪯ y, we have a x ⪯ a y.   (ii) For any sequences { x n }, { y n } ⊂ A with x n → x ( n → ∞ ) and y n → y ( n → ∞ ) where x, y ∈ A, we have x n y n → x y ( n → ∞ ).  .

Under the assumptions of Theorem  4.2, the following assertions hold true.

Then the mapping is well defined and the following assertions hold true.

Theorem 4.1 For problem (1.1), if 2 a 2 2 + 45 μ a 3 + 9 α a 3 < 0 is satisfied, then the following assertions hold true: (1) If λ ≤ μ + α, the steady state u = 0 is locally asymptotically stable.

The following assertions hold true.