Considera duos homines \(A\) et \(B\) qui non eundem natalem habent, ubi minor natu \(A\) est quam \(B\) . Ostende: Constellationes prorsus duae sunt aetatis \(a,b \in \mathbb{N}\) quibus haec applicatio: \(2\cdot a = b\) . Primum constituimus \(d \in \mathbb{R}^+\) ut aetatis differentiam inter \(A\) et \(B\) nascentem \(A\) cum \( d = d_0 + d_1 \) , \( d_0 \in \mathbb{N}_0, d_1 \in \mathbb{R}, d_1 \in [0;1[\) . Nunc consideramus tempus arbitrarium \(x \in \mathbb{R}^+\) post natum \(A\) cum \(x = x_0 + x_1\) , \(x_0 \in \mathbb{N}_0, x_1 \in \mathbb{R}, x_1 \in [0;1[\) .
At iam hoc tempore per definitionem: \(a = \lfloor x \rfloor \) et \(b = \lfloor x+d \rfloor\) . Nunc definiri \(x\) quam habet:
$$2 \lfloor x \rfloor = \lfloor x+d \rfloor \Leftrightarrow 2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor$$
1 si: \(0 \leq x_1 + d_1 < 1\):
Et $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 \Leftrightarrow x_0 = d_0.$$
Et hoc modo, quod $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + x_1 \rfloor = d_0$$ et $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ prima aetate $$b = \lfloor x + d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + \underbrace{x_1 + d_1}_{< 1} \rfloor = 2 d_0$$ pro.
2 si: \( 1 \leq x_1 + d_1 < 2 \):
Et $$2 \lfloor x_0 + x_1 \rfloor = \lfloor x_0 + d_0 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor \Leftrightarrow 2 x_0 = x_0 + d_0 + 1 \Leftrightarrow x_0 = d_0 + 1.$$
Et hoc modo, quod $$a = \lfloor x \rfloor = \lfloor x_0 + x_1 \rfloor = \lfloor d_0 + 1 + x_1 \rfloor = d_0 + 1$$ et $$b = \lfloor x+d \rfloor = \lfloor x_0 + x_1 + d_0 + d_1 \rfloor = \lfloor 2 d_0 + 1 + \underbrace{x_1 + d_1}_{\geq 1} \rfloor = 2 d_0 + 2$$ ad alterum constellatio desideravit aetatem.
Concretis verbis hoc significat, exempli gratia: Si mater tua te peperisset annos \(20\) , ea prorsus duplo nata est quam tu \(40\) et \(42\) . Casus num et quando mater tua est \(n\) temporibus antiquis, etiam interest: Hic poscis \(n \in \mathbb{N}\) ad placitum et posside \(x_0 = \frac{d_0}{n-1} \in \mathbb{N} \Leftrightarrow d_0 = k (n-1)\) . Hoc operatur si et tantum, si aevi integri differentia \( \lfloor d \rfloor = d_0 \) multiplex est e \(n-1\) . Verbi gratia, in casu superiore, mater tua annorum \(24\) \(6\) est aetas tua.