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

<b>set </b><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>;<br/>
<a NAME="E1:50_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p2"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">m</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span></span>)</span> = 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span>
<b>by </b><i><a class="ref" href="mcart_1.html#T39" title="MCART_1:th.39">MCART_1:39</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">m</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:50_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="altcat_1.html#T12" title="ALTCAT_1:th.12">ALTCAT_1:12</a></i>;<br/>
<a NAME="E6:50_1_1"/>
<font color="Maroon" title="c6">m</font> is    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> or <font color="Maroon" title="c6">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>) )</span><br/>

<b>assume </b><a NAME="E1:50_1_1_2"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="relset_1.html#M2" title="RELSET_1:mode.2">M5</a>(<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>)</span><br/>

<a NAME="E2:50_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T39" title="MCART_1:th.39">MCART_1:39</a></i>;<br/>
<a NAME="E3:50_1_1_2"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:50_1_1_2"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E26">A4</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="txt" href="#E3:50_1_1_2"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="#E5:50_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:50_1_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>
;<br/>
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="txt" href="#E3:50_1_1_2"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="txt" href="#E5:50_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:50_1_1_2"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="altcat_1.html#D5" title="ALTCAT_1:def.5">ALTCAT_1:def 5</a></i>
;<br/>
<b>hence </b><a NAME="E7:50_1_1_2"/>
<font color="Maroon" title="c6">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="#E3:50_1_1_2"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E5:50_1_1_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">m</font> = <font color="Maroon" title="c6">m</font> as   <a href="altcat_1.html#NM3" title="ALTCAT_1:NM.3">BinComp</a> of <font color="Maroon" title="c4">a</font> ;<br/>
<b>set </b><font color="Maroon" title="c8">D</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #);<br/>
<a NAME="E8:50_1_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) is    <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1_1_3"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D11" target="_self" title="ALTCAT_2:def.11">ALTCAT_2:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> )</span><br/>

<b>thus </b><a NAME="E2:50_1_1_3"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="altcat_2.html#R2" title="ALTCAT_2:pred.2">cc=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_1_3_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:50_1_1_3_1"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/>the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/>

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<b>assume </b><a NAME="E2:50_1_1_3_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E3:50_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></i>;<br/>
<a NAME="E4:50_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:50_1_1_3_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 <b>by </b><i><a class="txt" href="#E5:50_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E3:50_1_1_3"/>
<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D2" target="_self" title="ALTCAT_2:def.2">ALTCAT_2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span><br/>

<b>let </b><font color="Maroon" title="c9">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )</span><br/>

<b>assume </b><a NAME="E4:50_1_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #),the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #)</span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span><br/>

<a NAME="E5:50_1_1_3"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T39" title="MCART_1:th.39">MCART_1:39</a></i>;<br/>
<a NAME="E6:50_1_1_3"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then </b>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> = 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
<br/>.= 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
<b>by </b><i><a class="txt" href="#E6:50_1_1_3"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E8:50_1_1_3"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">D</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c7">m</font> #) as  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c9">D</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E10:50_1_1"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">:-&gt;</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c9">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
