<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:15_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="prob_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="prob_3.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E2:15_1"/>
<span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:15"/>
 <a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">c<sub>2</sub></font> is <a href="prob_1.html#V2">non-increasing</a>
 <b>by </b><i><a class="ref" href="prob_2.html#T14">PROB_2:14</a></i>;<br/>


</div>
