<?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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</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>4</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="ordinal1.html#K5">omega</a> ;<br/>







<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E1:29"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> is <a href="zf_fund1.html#V8" target="_self">closed_wrt_A1-A7</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:29"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="zf_fund1.html#V5" target="_self">closed_wrt_A5</a>
 
<b>by </b><i><a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="zf_fund1.html#V4" target="_self">closed_wrt_A4</a>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_fund1.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E5:29"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<a NAME="E1:29_1"/><i><font color="Green">E39</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funct_5.html#T64">FUNCT_5:64</a></i>;<br/>
<a NAME="E2:29_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_fund1.html#T3" target="_self">Th3</a></i>;<br/>
<b>hence </b><a NAME="E3:29_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_fund1.html#E1:29_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:29"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_1"><b>suppose </b></a><a NAME="E1:29_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T56">ZFMISC_1:56</a></i>;<br/><b>hence </b><a NAME="E3:29_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>]
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zfmisc_1.html#T46">ZFMISC_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_2_1_2"><b>suppose </b></a><a NAME="E1:29_2_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:29_2_1_2"/><i><font color="Green">E42</font></i>: 
 <a href="ordinal1.html#K5">omega</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="ref" href="zf_fund1.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:29_2_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green">E40</font></i></a></i>;<br/><a NAME="E4:29_2_1_2"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:29_2_1_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E5:29_2_1_2"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_fund1.html#E2:29_2_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29_2_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="zf_fund1.html#T2" target="_self">Th2</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <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#E4:29_2_1_2"><i><font color="Green">E44</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> ;<br/><a NAME="E7:29_2_1_2"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_2_1_2_1"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
  <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="29_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</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="E1:29_2_1_2_1_1"/>
<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"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:29_2_1_2_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E4:29_2_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:29_2_1_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:29_2_1_2_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<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>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</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#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:29_2_1_2_1_1"/>
<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>8</sub></font>,<font color="Maroon">c<sub>12</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#E3:29_2_1_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="grfunc_1.html#T18">GRFUNC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E9:29_2_1_2_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_1_1"><i><font color="Green">E48</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>10</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="E2:29_2_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><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> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:29_2_1_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>10</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="classes2.html#K6">]</a></span></span><a href="classes2.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</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#E4:29_2_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="grfunc_1.html#T15">GRFUNC_1:15</a></i>;<br/>
<a NAME="E6:29_2_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:29_2_1_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <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#E4:29_2_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T23">RELAT_1:23</a></i>;<br/>
<a NAME="E8:29_2_1_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>13</sub></font> <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#E4:29_2_1_2_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>hence </b><a NAME="E9:29_2_1_2_1"/>
<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"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:29_2_1_2_1"><i><font color="Green">E47</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="E8:29_2_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29"><i><font color="Green">E34</font></i></a>, <a class="txt" href="zf_fund1.html#E3:29"><i><font color="Green">E36</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29_2_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="zf_fund1.html#D9" target="_self">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</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#E1:29_2"><i><font color="Green">E40</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">c<sub>2</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#E8:29_2_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> ;<br/><a NAME="E11:29_2_1_2"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_2_1_2_2"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
  <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="29_2_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</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="E1:29_2_1_2_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:29_2_1_2_2_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>;<br/>
<i><font color="Green">E48</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E5:29_2_1_2_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E6:29_2_1_2_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <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#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:29_2_1_2_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</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#E1:29_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E10:29_2_1_2_2_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E11:29_2_1_2_2_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <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#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E12:29_2_1_2_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E9:29_2_1_2_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</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#E8:29_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zf_fund1.html#T1" target="_self">Th1</a></i>;<br/>
<font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="relat_1.html#K7">|</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T107">RELAT_1:107</a></i>
;<br/>
<a NAME="E15:29_2_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E3:29_2_1_2_2_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:29_2_1_2_2_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="zf_fund1.html#E7:29_2_1_2_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_fund1.html#E12:29_2_1_2_2_1"><i><font color="Green">E51</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>13</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="E2:29_2_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="classes2.html#K7">\/</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</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="E4:29_2_1_2_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:29_2_1_2_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</sub></font> <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#E4:29_2_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:29_2_1_2_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>17</sub></font> <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#E4:29_2_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c<sub>17</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="zf_fund1.html#E1:29_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="grfunc_1.html#T31">GRFUNC_1:31</a></i>;<br/>
<a NAME="E10:29_2_1_2_2"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="grfunc_1.html#T33">GRFUNC_1:33</a></i>;<br/>
<a NAME="E11:29_2_1_2_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>16</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>17</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="grfunc_1.html#T33">GRFUNC_1:33</a></i>;<br/>
<a NAME="E12:29_2_1_2_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="classes2.html#K7">\/</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<b>hence </b><a NAME="E13:29_2_1_2_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E4:29_2_1_2_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29_2_1_2_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="zf_fund1.html#E10:29_2_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E12:29_2_1_2"/>
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="zf_fund1.html#E2:29"><i><font color="Green">E35</font></i></a>, <a class="txt" href="zf_fund1.html#E1:29_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="zf_fund1.html#E8:29_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="zf_fund1.html#D10" target="_self">Def10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E7:29"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>from </b><i><a class="ref" href="finset_1.html#S2">FINSET_1:sch 2</a>(<a class="txt" href="zf_fund1.html#E4:29"><i><font color="Green">E37</font></i></a>, <a class="txt" href="zf_fund1.html#E5:29"><i><font color="Green">E38</font></i></a>, <a class="txt" href="zf_fund1.html#E6:29"><i><font color="Green">E39</font></i></a>);</i><br/>


</div>
