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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/>













<b>thus </b><a NAME="E1:5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> <a href="bvfunc_1.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<i><font color="Green">E6</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
;<br/>
<a NAME="E2:5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_6.html#T39">BVFUNC_6:39</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> <a href="bvfunc_1.html#R1">'&lt;'</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_9.html#E1:5_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bvfunc_1.html#T19">BVFUNC_1:19</a></i>;<br/>


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

<i><font color="Green">E6</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="bvfunc_1.html#T7">BVFUNC_1:7</a></i>
;<br/>
<a NAME="E3:5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="bvfunc_1.html#K14">'imp'</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bvfunc_1.html#K19">I_el</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="bvfunc_6.html#T39">BVFUNC_6:39</a></i>;<br/>
<b>hence </b><a NAME="E4:5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="valuat_1.html#K6">'&amp;'</a> <font color="Maroon">c<sub>7</sub></font> <a href="bvfunc_1.html#R1">'&lt;'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="bvfunc_9.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="bvfunc_1.html#T19">BVFUNC_1:19</a></i>;<br/>


</div>
