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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font>, <font color="Maroon" title="c4">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c5">M1</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/><b>let </b><font color="Maroon" title="c6">M2</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c5">M1</font> is <a href="altcat_3.html#V9" title="ALTCAT_3:attr.9">_zero</a>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c6">M2</font> is <a href="altcat_3.html#V9" title="ALTCAT_3:attr.9">_zero</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D12" target="_self" title="ALTCAT_3:def.12">ALTCAT_3:def 12</a><br/>

<b>assume </b><a NAME="E3:51"/><i><font color="Green" title="E31">A3</font></i>: 
<font color="Maroon" title="c7">o</font> is <a href="altcat_3.html#V8" title="ALTCAT_3:attr.8">_zero</a>
 ;<br/>

<a NAME="E4:51"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c7">o</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> &amp; <font color="Maroon" title="c7">o</font> is <a href="altcat_3.html#V7" title="ALTCAT_3:attr.7">terminal</a> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D11" target="_self" title="ALTCAT_3:def.11">Def11</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">A</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font>;<br/><b>let </b><font color="Maroon" title="c9">B</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font>;<br/>



<b>consider </b><font color="Maroon" title="c10">B1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">o2</font><b> such that </b><br/><a NAME="E6:51"/><i><font color="Green" title="E33">A5</font></i>: 
( <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">B2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font><b> such that </b><br/><a NAME="E8:51"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">A1</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font><b> such that </b><br/><a NAME="E10:51"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">o1</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self" title="ALTCAT_3:def.10">Def10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">A2</font> being   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font><b> such that </b><br/><a NAME="E12:51"/><i><font color="Green" title="E36">A8</font></i>: 
( <font color="Maroon" title="c13">A2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self" title="ALTCAT_3:def.10">Def10</a></i>;<br/>
<a NAME="E13:51"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c6">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">A2</font>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E2:51"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="altcat_3.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="altcat_3.html#D12" target="_self" title="ALTCAT_3:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:51"/><i><font color="Green" title="E38">A10</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E16:51"/><i><font color="Green" title="E39">A11</font></i>: 
( <font color="Maroon" title="c8">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> &amp; <font color="Maroon" title="c12">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E15:51"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:51"/><i><font color="Green" title="E40">A12</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E19:51"/><i><font color="Green" title="E41">A13</font></i>: 
( <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> &amp; <font color="Maroon" title="c11">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E18:51"><i><font color="Green" title="E40">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:51"/>
 ex <font color="Olive" title="b1">M</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font> st <br/>( <font color="Olive" title="b1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="altcat_3.html#D10" target="_self" title="ALTCAT_3:def.10">Def10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E22:51"/><i><font color="Green" title="E42">A14</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E23:51"/><i><font color="Green" title="E43">A15</font></i>: 
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z</font> &amp; <font color="Maroon" title="c13">A2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E22:51"><i><font color="Green" title="E42">A14</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E24:51"/><i><font color="Green" title="E44">A16</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E6:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="altcat_3.html#E10:51"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="altcat_3.html#E12:51"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c6">M2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">M1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">B2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">A1</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_3.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="altcat_3.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="altcat_3.html#E13:51"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="altcat_3.html#D12" target="_self" title="ALTCAT_3:def.12">Def12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">B2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">A2</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">B1</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c12">A1</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E24:51"><i><font color="Green" title="E44">A16</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">o</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E6:51"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="altcat_3.html#E12:51"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="altcat_3.html#E16:51"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="altcat_3.html#E19:51"><i><font color="Green" title="E41">A13</font></i></a>, <a class="txt" href="altcat_3.html#E23:51"><i><font color="Green" title="E43">A15</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon" title="c9">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">A</font>
<b>by </b><i><a class="txt" href="altcat_3.html#E8:51"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
;<br/>


</div>
