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

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>4</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">a<sub>5</sub></font>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>)
 ;<br/><a NAME="E2:75_1_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:75_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>2</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="funct_1.html#NM1">Function</a>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:75_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:75_1_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E3:75_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E1:75"><i><font color="Green">E37</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_2"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:75_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) &amp; <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) &amp; <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font>),<font color="Olive">b<sub>7</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font>))
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:75_1_3"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>) &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="zf_fund1.html#K1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
;<br/>
<b>hence </b><a NAME="E4:75_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>),<font color="Maroon">c<sub>7</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>))
 <b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:75_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>take </b><font color="Maroon">c<sub>2</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>);<br/>

<b>thus </b><a NAME="E1:75_1_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:75_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>)
 ;<br/>

<a NAME="E3:75_1_4"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:75_1_4"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_4"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>2</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_4"><i><font color="Green">E45</font></i></a>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E5:75_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>take </b><font color="Maroon">c<sub>2</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>);<br/>

<b>thus </b><a NAME="E1:75_1_5"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green">E39</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:75_1_5"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>)
 ;<br/>

<a NAME="E3:75_1_5"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:75_1_5"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>1</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_5"><i><font color="Green">E45</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_5"><i><font color="Green">E46</font></i></a>, <a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_5"><i><font color="Green">E46</font></i></a>, <a class="ref" href="relat_1.html#T79">RELAT_1:79</a></i>
;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being  <a href="altcat_1.html#V6">strict</a> <a href="altcat_1.html#NM4">category</a><b> such that </b><br/><a NAME="E7:75_1"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E8:75_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>and </b><br/><a NAME="E9:75_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <font color="Olive">b<sub>5</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S4" target="_self">YELLOW18:sch 4</a>(<a class="txt" href="yellow18.html#E2:75_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="yellow18.html#E3:75_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="yellow18.html#E4:75_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="yellow18.html#E5:75_1"><i><font color="Green">E44</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM9">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:75_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E12:75_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow18.html#V2" target="_self">para-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_6"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7" target="_self">YELLOW18:def 7</a><br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E1:75_1_6"/>
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>) &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:75_1_6"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:75_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V9">semi-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D14">ALTCAT_1:def 14</a><br/>

<b>assume </b><a NAME="E1:75_1_7"/><i><font color="Green">E51</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>





<b>assume </b><a NAME="E2:75_1_7"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<b>hence </b><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>8</sub></font> <a href="zf_fund1.html#K1">(#)</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E9:75_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_7"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="yellow16.html#T1">YELLOW16:1</a></i>
;<br/><br/>


</div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:75_1_8"/>
(  <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>) &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green">E39</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green">E46</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:75_1_8_1"/><i><font color="Green">E52</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E2:75_1_8_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>) <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green">E38</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:75_1_8_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1_8_1"><i><font color="Green">E52</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:75_1_8_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>) &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>5</sub></font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_8_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>6</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>)</span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E13:75_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_8_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="yellow18.html#E5:75_1_8_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="altcat_1.html#D14">ALTCAT_1:def 14</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_8_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="relat_1.html#T78">RELAT_1:78</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:75_1_8"/>
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 <b>by </b><i><a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:75_1_9"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>thus </b><font color="Maroon">c<sub>1</sub></font> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow18.html#D8" target="_self">Def8</a></i>
<br/>.= 
 <a href="funct_5.html#K1">proj1</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>)</span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E14:75_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>)</span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
<b>by </b><i><a class="ref" href="relat_1.html#T71">RELAT_1:71</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green">E48</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E16:75_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="yellow18.html#V3" target="_self">set-id-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D10" target="_self">YELLOW18:def 10</a><br/>

<b>thus </b> <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="partfun1.html#K6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E14:75_1"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E15:75_1"><i><font color="Green">E52</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  <a href="altcat_1.html#V6">strict</a> <a href="altcat_1.html#V9">semi-functional</a> <a href="yellow18.html#V2" target="_self">para-functional</a> <a href="yellow18.html#V3" target="_self">set-id-inheriting</a> <a href="altcat_1.html#NM4">category</a> <b>by </b><i><a class="txt" href="yellow18.html#E12:75_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow18.html#E13:75_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>thus </b><a NAME="E18:75_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green">E45</font></i></a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b> <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E15:75_1"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green">E48</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E20:75_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font> holds <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green">E46</font></i></a></i>;<br/>


</div>
