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<b>let </b><font color="Maroon">C1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">C2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<a NAME="E1:27"/><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</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">C1</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</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">C2</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">C1</font>,<font color="Maroon">C2</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/><b>hence </b><a NAME="E2:27_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>, <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>, <font color="Maroon">c'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>;<br/><b>assume </b><a NAME="E3:27_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/><a NAME="E4:27_1"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>thus </b><i><font color="Green">E43</font></i>: <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="altcat_2.html#K1">~</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
<b>by </b><i>, , </i>
;<br/><i><font color="Green">E48</font></i>: <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="altcat_1.html#K1">^&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="altcat_2.html#K1">~</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
<b>by </b><i>, , </i>
;<br/><a NAME="E7:27_1"/><i><font color="Green">E49</font></i>: 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c'</font>,<font color="Maroon">b'</font>,<font color="Maroon">a'</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>assume </b><a NAME="E8:27_1"/><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">c</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">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">c</font>;<br/><a NAME="E9:27_1"/>
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E10:27_1"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:27_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">g</font>,<font color="Maroon">f</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">a'</font>;<br/><b>let </b><font color="Maroon">g'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c'</font>,<font color="Maroon">b'</font>;<br/><b>assume </b><a NAME="E12:27_1"/>
( <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> &amp; <font color="Maroon">g'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g'</font> = 
<span class="p1">(<span class="default"><a href="functor0.html#K1">~</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i>, , , , <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g</font>,<font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E3:27"/><i><font color="Green">E54</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font><br/> for <font color="Olive">a'</font>, <font color="Olive">b'</font>, <font color="Olive">c'</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>  st <font color="Olive">a'</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <font color="Olive">b'</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">c'</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> holds <br/>( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b'</font>,<font color="Olive">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</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</font>,<font color="Olive">c</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>  implies for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">a</font>,<font color="Olive">b</font><br/> for <font color="Olive">g</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b</font>,<font color="Olive">c</font><br/> for <font color="Olive">f'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">b'</font>,<font color="Olive">a'</font><br/> for <font color="Olive">g'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">c'</font>,<font color="Olive">b'</font>  st <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">g'</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> holds <br/><font color="Olive">f'</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">g'</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">f</font> ) )
 ;<br/>

<b>thus </b><a NAME="E5:27"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D3" target="_self">YELLOW18:def 3</a><br/>

<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E4:27_2_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:27_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> )
 ;<br/><a NAME="E3:27_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E4:27_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">y</font>, <font color="Maroon">b</font> = <font color="Maroon">z</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font> <b>by </b><i/>;<br/><b>thus </b>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span>

<br/>.= 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>
<b>by </b><i/>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:27"/>
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a>  <a href="altcat_2.html#K1">~</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">c</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>;<br/><b>let </b><font color="Maroon">a'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>, <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>, <font color="Maroon">c'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>;<br/>



<b>assume </b><a NAME="E9:27"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">c'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E10:27"/><i><font color="Green">E67</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">c'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">c'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:27"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</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</font>,<font color="Maroon">b</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="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E12:27"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E13:27"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E14:27"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</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</font>,<font color="Maroon">b</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="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span>,<span class="p2"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  implies <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">a</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="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E69</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E106</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:27_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:27_4_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y</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">b'</font>,<font color="Maroon">c'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">z</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">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E4:27_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:27_4"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</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">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> )
 ;<br/><a NAME="E3:27_4"/>
( <font color="Maroon">z</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">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">y</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</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>hence </b><a NAME="E4:27_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:27_5"/><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>
 ;<br/><a NAME="E2:27_5"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">y</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">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <font color="Maroon">z</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</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">y</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b</font>,<font color="Maroon">a</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">z</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c</font>,<font color="Maroon">b</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">y</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <b>by </b><i>, , </i>;<br/><b>reconsider </b><font color="Maroon">g'</font> = <font color="Maroon">z</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">c'</font> <b>by </b><i>, , </i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">g'</font>,<font color="Maroon">f'</font>

<br/>.= 
<font color="Maroon">g'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font>
<b>by </b><i>, , <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">g</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">g</font>
<b>by </b><i>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E19:27"/>
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="hidden.html#R1">=</a>  <a href="functor0.html#K1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">c</font>,<font color="Maroon">b</font>,<font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>


</div>
