<?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> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:93_1_2"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="yellow18.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E2:93_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) is   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow18.html#D12" target="_self">Def12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<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> ] means ex <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>ex <font color="Olive">b<sub>3</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>ex <font color="Olive">b<sub>4</sub></font> being  <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>5</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font> st <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</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/><font color="Olive">b<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>6</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>6</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> ) );<br/>
<a NAME="E4:93_1_2"/><i><font color="Green">E47</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<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>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_1"><b>proof </b></a><div class="add">

<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="E1:93_1_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:93_1_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>6</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> &amp; <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E3:93_1_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<b>assume </b><a NAME="E5:93_1_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E6:93_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:93_1_2_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E7:93_1_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>5</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>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 
;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>9</sub></font><b> such that </b><br/><a NAME="E10:93_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E11:93_1_2_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</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="Maroon">c<sub>8</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/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E7:93_1_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="yellow18.html#T46" target="_self">Th46</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>11</sub></font>
;<br/>

<b>thus </b><a NAME="E12:93_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>] )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:93_1_2_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="yellow18.html#E10:93_1_2_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yellow18.html#E11:93_1_2_1"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="pboole.html#M3">ManySortedFunction</a> of the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E6:93_1_2"/><i><font color="Green">E48</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> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font>] )
 <b>from </b><i><a class="ref" href="yellow18.html#S23" target="_self">YELLOW18:sch 23</a>(<a class="txt" href="yellow18.html#E4:93_1_2"><i><font color="Green">E47</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <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>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>3</sub></font>;<br/>
<a NAME="E7:93_1_2"/><i><font color="Green">E49</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> 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>  holds <br/>for <font color="Olive">b<sub>3</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> holds <font color="Maroon">H<sub>2</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>) <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_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="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:93_1_2_2"/><i><font color="Green">E50</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</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>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>

<a NAME="E2:93_1_2_2"/><i><font color="Green">E51</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>thus </b><a NAME="E3:93_1_2_2"/>
<font color="Maroon">H<sub>2</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#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>5</sub></font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="yellow18.html#E1:93_1_2_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_2"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:93_1_2"/><i><font color="Green">E50</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><br/>for <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) &amp; <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>2</sub></font>) &amp; <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>) holds <br/>for <font color="Olive">b<sub>9</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>7</sub></font><br/>for <font color="Olive">b<sub>10</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>8</sub></font> st <font color="Olive">b<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</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>) &amp; <font color="Olive">b<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</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>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<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="Olive">b<sub>10</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_3"><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>, <font color="Maroon">c<sub>7</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="E1:93_1_2_3"/><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>5</sub></font>,<font color="Maroon">c<sub>6</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>6</sub></font>,<font color="Maroon">c<sub>7</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>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E2:93_1_2_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>;<br/>



<b>assume </b><a NAME="E3:93_1_2_3"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<a NAME="E4:93_1_2_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>16</sub></font><b> such that </b><br/><a NAME="E6:93_1_2_3"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 <b>and </b><br/><a NAME="E7:93_1_2_3"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</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="Maroon">c<sub>15</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/><font color="Maroon">c<sub>18</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E8:93_1_2_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>19</sub></font>, <font color="Maroon">c<sub>20</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>21</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>22</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>19</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>20</sub></font><b> such that </b><br/><a NAME="E10:93_1_2_3"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>and </b><br/><a NAME="E11:93_1_2_3"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>19</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="Maroon">c<sub>19</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/><font color="Maroon">c<sub>22</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E12:93_1_2_3"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</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> 
 
<b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E13:93_1_2_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E12:93_1_2_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>25</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>, <font color="Maroon">c<sub>26</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>23</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>24</sub></font><b> such that </b><br/><a NAME="E15:93_1_2_3"/><i><font color="Green">E61</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>and </b><br/><a NAME="E16:93_1_2_3"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</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="Maroon">c<sub>23</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/><font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>23</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>25</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>24</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E17:93_1_2_3"/><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</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>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="yellow18.html#E4:93_1_2_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="yellow18.html#E8:93_1_2_3"><i><font color="Green">E57</font></i></a></i>;<br/>
<a NAME="E18:93_1_2_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E10:93_1_2_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="yellow18.html#E15:93_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>18</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>6</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>22</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E18:93_1_2_3"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>29</sub></font> = <font color="Maroon">c<sub>26</sub></font> as   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>5</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E18:93_1_2_3"><i><font color="Green">E64</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>32</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E2:93_1_2_3_1"/><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>31</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; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>32</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="yellow18.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E3:93_1_2_3_1"/><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>6</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> 
 <b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>30</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E15:93_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="yellow18.html#E16:93_1_2_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="yellow18.html#E18:93_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_3_1"><i><font color="Green">E65</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>32</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_3_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>32</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E10:93_1_2_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="yellow18.html#E11:93_1_2_3"><i><font color="Green">E59</font></i></a>, <a class="txt" href="yellow18.html#E18:93_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="yellow18.html#E3:93_1_2_3_1"><i><font color="Green">E66</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E7:93_1_2_3"><i><font color="Green">E56</font></i></a>, <a class="txt" href="yellow18.html#E18:93_1_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="yellow18.html#E2:93_1_2_3_1"><i><font color="Green">E65</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>27</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>30</sub></font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E23:93_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E24:93_1_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E3:93_1_2_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E6:93_1_2_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E10:93_1_2_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="yellow18.html#E15:93_1_2_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="yellow18.html#E17:93_1_2_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="yellow18.html#T38" target="_self">Th38</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:93_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>, <a href="altcat_1.html#K8">idm</a> <font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_4"><b>proof </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>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:93_1_2_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E2:93_1_2_4"/>
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E4:93_1_2_4"/><i><font color="Green">E53</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E5:93_1_2_4"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</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="Maroon">c<sub>7</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/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E6:93_1_2_4"/><i><font color="Green">E55</font></i>: 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <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>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow18.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E7:93_1_2_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E4:93_1_2_4"><i><font color="Green">E53</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>6</sub></font>;<br/><a NAME="E1:93_1_2_4_1"/><i><font color="Green">E57</font></i>: 
 <a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_4"><i><font color="Green">E52</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:93_1_2_4_1"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</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; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="yellow18.html#T45" target="_self">Th45</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E4:93_1_2_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E5:93_1_2_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="yellow18.html#E7:93_1_2_4"><i><font color="Green">E56</font></i></a>, <a class="txt" href="yellow18.html#E3:93_1_2_4_1"><i><font color="Green">E58</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E3:93_1_2_4_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p2">(<span class="default"><a href="yellow18.html#K3" target="_self">the_carrier_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_4_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:93_1_2_4"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:93_1_2_4"><i><font color="Green">E52</font></i></a>, <a class="txt" href="yellow18.html#E4:93_1_2_4"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E6:93_1_2_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E7:93_1_2_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being  <a href="functor0.html#V9">strict</a> <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>1</sub></font>, <a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:93_1_2"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="functor0.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>and </b><br/><a NAME="E12:93_1_2"/><i><font color="Green">E53</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> 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>  holds <br/>for <font color="Olive">b<sub>3</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> holds <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</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>)
 <b>from </b><i><a class="ref" href="yellow18.html#S8" target="_self">YELLOW18:sch 8</a>(<a class="txt" href="yellow18.html#E2:93_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="yellow18.html#E7:93_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="yellow18.html#E8:93_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="yellow18.html#E9:93_1_2"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E13:93_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="yellow18.html#E11:93_1_2"><i><font color="Green">E52</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E14:93_1_2"/><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</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>8</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>

<a NAME="E15:93_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E14:93_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="yellow18.html#K4" target="_self">Concretized</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="yellow18.html#K3" target="_self">-carrier_of</a> <font color="Maroon">c<sub>10</sub></font><b> such that </b><br/><a NAME="E17:93_1_2"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E18:93_1_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</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="Maroon">c<sub>9</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/><font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E6:93_1_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E19:93_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="yellow18.html#E12:93_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="yellow18.html#E14:93_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="yellow18.html#E17:93_1_2"><i><font color="Green">E55</font></i></a></i>;<br/>
<a NAME="E20:93_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</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>  )
 
<b>by </b><i><a class="txt" href="yellow18.html#E17:93_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K6">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p3">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E17:93_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="yellow18.html#E18:93_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="yellow18.html#E19:93_1_2"><i><font color="Green">E57</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="yellow18.html#E14:93_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/><br/>


</div>
