<?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> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E47</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> )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V4" target="_self">closed_wrt_A4</a>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:43"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="zf_fund1.html#V1" target="_self">closed_wrt_A1</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E4:43"/><i><font color="Green">E50</font></i>: 
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T63">ZF_LANG1:63</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green">E51</font></i>: 
 <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T64">ZF_LANG1:64</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E7:43"/><i><font color="Green">E53</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T7" target="_self">Th7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</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> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:43"><i><font color="Green">E47</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> &amp;  <a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43"><i><font color="Green">E51</font></i></a>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><a NAME="E3:43_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="zf_fund1.html#K4" target="_self">x".</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#E7:43"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:43_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E5:43_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</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#E2:43_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></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>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> ;<br/><a NAME="E6:43_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></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>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></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>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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="43_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></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="E1:43_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></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>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</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="E3:43_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>
<a NAME="E4:43_1_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:43_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<a NAME="E5:43_1_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <a href="zf_lang.html#K1">VAR</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>
<br/>.= 
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>12</sub></font></span><a href="classes2.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:43_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>
;<br/>
<a NAME="E8:43_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p0">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T18">GRFUNC_1:18</a></i>;<br/>
<a NAME="E9:43_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E10:43_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<i><font color="Green">E61</font></i>: <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zf_lang1.html#T1">ZF_LANG1:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T1">ZF_LANG1:1</a></i>
;<br/>
<a NAME="E12:43_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E13:43_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zf_model.html#T7">ZF_MODEL:7</a></i>;<br/>
<b>hence </b><a NAME="E14:43_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E10:43_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>11</sub></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="E2:43_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:43_1_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="zf_fund1.html#K4" target="_self">x".</a> <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#E3:43_1_1"><i><font color="Green">E56</font></i></a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> = 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>
<br/>.= 
<span class="p1"><a href="classes2.html#K1">{</a><span class="default"><font color="Maroon">c<sub>13</sub></font></span><a href="classes2.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:43_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>
;<br/>
<a NAME="E7:43_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p5">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p0">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T18">GRFUNC_1:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:43_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="classes2.html#K1">{</a><span class="default"><span class="p2"><a href="classes2.html#K6">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E4:43_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_fund1.html#E7:43_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E10:43_1_1_1"/>
<font color="Maroon">c<sub>13</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>13</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/>
<a NAME="E11:43_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E2:43_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="zf_fund1.html#E19"><i><font color="Green">Lemma18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E12:43_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="classes2.html#K1">{</a><span class="default"><span class="p3"><a href="classes2.html#K6">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></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>  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E9:43_1_1_1"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:43_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E2:43"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_fund1.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E8:43_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_2"/><i><font color="Green">E58</font></i>: 
not  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:43_1_1_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:43_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E6:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_model.html#T8">ZF_MODEL:8</a></i>;<br/>
<a NAME="E7:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<a NAME="E8:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_1"><i><font color="Green">E54</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:43_1_1"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_1_2_1"/>
 <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then </b><font color="Maroon">c<sub>4</sub></font> = 
 <a href="zf_lang.html#K2">x.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="zf_fund1.html#D3" target="_self">Def3</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="zf_fund1.html#D3" target="_self">Def3</a></i>
;<br/><b>hence </b><a NAME="E3:43_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/><a NAME="E3:43_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E20"><i><font color="Green">Lemma19</font></i></a></i>;<br/><a NAME="E4:43_1_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:43_1_2_2"/><i><font color="Green">E57</font></i>: 
(  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</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#E7:43"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E2:43_1_2_2"/>
(  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E37"><i><font color="Green">Lemma36</font></i></a></i>;<br/>
<a NAME="E3:43_1_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E1:43_1_2_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:43_1_2_2"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where 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> <font color="Maroon">c<sub>8</sub></font> </span>}</span> ;<br/><a NAME="E6:43_1_2"/><i><font color="Green">E57</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where 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> <font color="Maroon">c<sub>8</sub></font> </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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T17" target="_self">Th17</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> where 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> <font color="Maroon">c<sub>8</sub></font> </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span> ;<br/><a NAME="E8:43_1_2"/><i><font color="Green">E58</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43"><i><font color="Green">E49</font></i></a>, <a class="ref" href="zf_fund1.html#D6" target="_self">Def6</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K2">{</a><span class="default"><span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p3"><a href="tarski.html#K4">[</a><span class="default"><span class="p4">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></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> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> ) </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>12</sub></font> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>14</sub></font> </span>}</span> ;<br/><a NAME="E10:43_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<a href="ordinal1.html#K5">omega</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="zf_fund1.html#E2:43_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="grfunc_1.html#T19">GRFUNC_1:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>, <font color="Maroon">c<sub>19</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="grfunc_1.html#T15">GRFUNC_1:15</a></i>;<br/>
<a NAME="E3:43_1_2_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>;<br/>
<b>then </b><i><font color="Green">E61</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="grfunc_1.html#T33">GRFUNC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></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"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>
<br/>.= 
 <a href="zf_fund1.html#K5" target="_self">code</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>
<i><font color="Green">E62</font></i>:  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="grfunc_1.html#T33">GRFUNC_1:33</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41">ENUMSET1:41</a></i>
;<br/>
<a NAME="E6:43_1_2_3"/>
1 <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E7:43_1_2_3"/><b>then </b>
(  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E8:43_1_2_3"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E5:43_1_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>hence </b><a NAME="E9:43_1_2_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>,<a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:43_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/><a NAME="E12:43_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>12</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1_2_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>12</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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="43_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></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="E1:43_1_2_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>12</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</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="E3:43_1_2_4_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>19</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="E5:43_1_2_4_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2_4_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E6:43_1_2_4_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43_1_2_4_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E46"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <a href="zf_lang.html#K1">VAR</a>  <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E5:43_1_2_4_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E8:43_1_2_4_1"/><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<i><font color="Green">E65</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E11:43_1_2_4_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43_1_2_4_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_fund1.html#E8:43_1_2_4_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E9:43_1_2_4_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_fund1.html#E47"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E12:43_1_2_4_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<font color="Maroon">c<sub>19</sub></font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E14:43_1_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E12:43_1_2_4_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="zf_model.html#T7">ZF_MODEL:7</a></i>;<br/>
<b>hence </b><a NAME="E15:43_1_2_4_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_2_4_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>17</sub></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="E2:43_1_2_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:43_1_2_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:43_1_2_4"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E6:43_1_2_4"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR1" target="_self">is_equality</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E7:43_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zf_model.html#T7">ZF_MODEL:7</a></i>;<br/>
<b>then </b><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="zf_lang1.html#T1">ZF_LANG1:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="zf_lang1.html#T1">ZF_LANG1:1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E10:43_1_2_4"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E11:43_1_2_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E8:43_1_2_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<a NAME="E12:43_1_2_4"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_2_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_2_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E47"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E13:43_1_2_4"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> 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#E6:43_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E15:43_1_2_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="zf_fund1.html#E12:43_1_2_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_fund1.html#E46"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:43_1_2_4"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>12</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E13:43_1_2_4"><i><font color="Green">E66</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E13:43_1_2"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E6:43_1_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_fund1.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E14:43_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>14</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1_2_5"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>14</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></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="43_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>17</sub></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="E1:43_1_2_5_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</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="E3:43_1_2_5_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</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="E5:43_1_2_5_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:43_1_2_5_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E6:43_1_2_5_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43_1_2_5_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_2_5_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E46"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>20</sub></font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="zf_fund1.html#E5:43_1_2_5_1"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E10:43_1_2_5_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="zf_lang.html#K1">VAR</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>23</sub></font> holds <br/><font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="funct_2.html#S6">FUNCT_2:sch 6</a>();<br/></i>
<a NAME="E11:43_1_2_5_1"/><i><font color="Green">E65</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E12:43_1_2_5_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2_5_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>20</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2_5_1"><i><font color="Green">E64</font></i></a></i>
;<br/>
<a NAME="E14:43_1_2_5_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43_1_2_5_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_fund1.html#E11:43_1_2_5_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="zf_fund1.html#E12:43_1_2_5_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="zf_fund1.html#E47"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E15:43_1_2_5_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E16:43_1_2_5_1"/>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:43_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_2_5_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2_5_1"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E17:43_1_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<a NAME="E18:43_1_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<a NAME="E19:43_1_2_5_1"/><b>then </b>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E15:43_1_2_5_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="zf_model.html#T8">ZF_MODEL:8</a></i>;<br/>
<b>hence </b><a NAME="E20:43_1_2_5_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E14:43_1_2_5_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>17</sub></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="E2:43_1_2_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="zf_lang.html#K1">VAR</a> ,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:43_1_2_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="zf_model.html#K5">St</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="zf_fund1.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E5:43_1_2_5"/><i><font color="Green">E62</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43"><i><font color="Green">E50</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43"><i><font color="Green">E51</font></i></a>, <a class="txt" href="zf_fund1.html#E3:43_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="zf_fund1.html#E16"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<a NAME="E6:43_1_2_5"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="partfun1.html#K1">*</a> <a href="zf_fund1.html#K3" target="_self">decode</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="zf_fund1.html#K5" target="_self">code</a> <span class="p3">(<span class="default"><a href="zf_fund1.html#K6" target="_self">Free</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:43"><i><font color="Green">E52</font></i></a>, <a class="txt" href="zf_fund1.html#E23"><i><font color="Green">Lemma22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E9:43_1_2_5"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><span class="p3">(<span class="default"><a href="zf_fund1.html#K4" target="_self">x".</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_2_5"><i><font color="Green">E61</font></i></a>, <a class="txt" href="zf_fund1.html#E5:43_1_2_5"><i><font color="Green">E62</font></i></a>, <a class="txt" href="zf_fund1.html#E6:43_1_2_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="zf_fund1.html#E47"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E10:43_1_2_5"/>
<font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font> <a href="zf_lang.html#NR2" target="_self">is_membership</a> 
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E11:43_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:43_1_2_5"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zf_model.html#T8">ZF_MODEL:8</a></i>;<br/>
<a NAME="E12:43_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<a NAME="E13:43_1_2_5"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T2">ZF_LANG1:2</a></i>;<br/>
<a NAME="E14:43_1_2_5"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3">(<span class="default"><a href="ordinal4.html#K3">1-element_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> 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#E8:43_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E16:43_1_2_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E11:43_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="zf_fund1.html#E9:43_1_2_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="zf_fund1.html#E46"><i><font color="Green">Lemma44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:43_1_2_5"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="zf_fund1.html#K2" target="_self">(#)</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where 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> <font color="Maroon">c<sub>14</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E14:43_1_2_5"><i><font color="Green">E65</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E15:43_1_2"/>
 <a href="zf_fund1.html#K7" target="_self">Diagram</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>3</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#E1:43"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E8:43_1_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="zf_fund1.html#E10:43_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="zf_fund1.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
