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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>



<b>thus </b><a NAME="E1:19"/>
 <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<a NAME="E2:19_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">A1</font>
 ;<br/>

<a NAME="E3:19"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:19_2"/>
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> holds <br/><font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:19_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E2:19_2_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> )
 <b>by </b><i>, <a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:19"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="prob_3.html#K1" target="_self">Partial_Intersection</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T29">PROB_1:29</a></i>;<br/>


</div>
