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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E16</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:16"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T11">FINSEQ_3:11</a></i>;<br/>
<a NAME="E3:16"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="card_2.html#T54">CARD_2:54</a></i>;<br/>
<a NAME="E4:16"/><b>then </b><i><font color="Green">E17</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T78">FINSEQ_1:78</a></i>;<br/>
<a NAME="E5:16"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E1:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E7:16"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:16"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E10:16"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T19">FINSEQ_3:19</a></i>;<br/>
<a NAME="E11:16"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="calcul_1.html#E9:16"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E12:16"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="calcul_1.html#E4:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a></i>;<br/>


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