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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="classes2.html#NM1">Universe</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="zf_refle.html#NM1">Subclass</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E4:24"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V5" target="_self">closed_wrt_A5</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V6" target="_self">closed_wrt_A6</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E7:24"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V7" target="_self">closed_wrt_A7</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E8:24"/><i><font color="Green">E34</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_fund1.html#E5:24"><i><font color="Green">E31</font></i></a>, <a class="ref" href="zf_fund1.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_1_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 ;<br/><a NAME="E4:24_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_1_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:24_1_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_1_2"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  iff <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E8:24"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:24"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K9">\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E12:24"/><i><font color="Green">E35</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K9">\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_fund1.html#E6:24"><i><font color="Green">E32</font></i></a>, <a class="ref" href="zf_fund1.html#D11" target="_self">Def11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K9">\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="classes2.html#K9">\</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 ;<br/><a NAME="E4:24_2_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K9">\</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_2_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K9">\</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:24_2_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K9">\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_2_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K9">\</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  iff <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K9">\</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_2"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="classes2.html#K9">\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E12:24"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E15:24"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> ;<br/>
<a NAME="E16:24"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:24"><i><font color="Green">E30</font></i></a>, <a class="txt" href="zf_fund1.html#E7:24"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zf_fund1.html#D12" target="_self">Def12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:24_3_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 ;<br/><a NAME="E4:24_3_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:24_3_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:24_3_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_3_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:24_3_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:24_3_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_3_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:24_3"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="classes2.html#K1">}</a></span> ) </span>}</span>  iff <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24_3"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E18:24"/><b>then </b>
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="classes2.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E16:24"><i><font color="Green">E36</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E19:24"/>
<font color="Maroon">c<sub>3</sub></font> <a href="zf_fund1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:24"><i><font color="Green">E29</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>


</div>
