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<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o3</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">M1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">M2</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o3</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:51"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">M1</font> is <a href="altcat_3.html#V9" target="_self">_zero</a>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">M2</font> is <a href="altcat_3.html#V9" target="_self">_zero</a>
 ;<br/>

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

<b>assume </b><a NAME="E3:51"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">o</font> is <a href="altcat_3.html#V8" target="_self">_zero</a>
 ;<br/>

<a NAME="E4:51"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">o</font> is <a href="altcat_3.html#V6" target="_self">initial</a> &amp; <font color="Maroon">o</font> is <a href="altcat_3.html#V7" target="_self">terminal</a> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E4:51"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o3</font>;<br/>



<b>consider </b><font color="Maroon">B1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o2</font><b> such that </b><br/><a NAME="E6:51"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">B1</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">o</font>,<font color="Maroon">o2</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">o</font>,<font color="Maroon">o2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">B2</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o3</font><b> such that </b><br/><a NAME="E8:51"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">B2</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">o</font>,<font color="Maroon">o3</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">o</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o</font><b> such that </b><br/><a NAME="E10:51"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">A1</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">o1</font>,<font color="Maroon">o</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">o1</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">A2</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o</font><b> such that </b><br/><a NAME="E12:51"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">A2</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">o2</font>,<font color="Maroon">o</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">o2</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<a NAME="E13:51"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">M2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="altcat_3.html#E6:51"><i><font color="Green">E28</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:51"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="altcat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E16:51"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:51"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o3</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="altcat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E19:51"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:51"/>
 ex <font color="Olive">M</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o</font> st <br/>( <font color="Olive">M</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">o</font>,<font color="Maroon">o</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">o</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span> is <a href="realset1.html#V1">trivial</a> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E10:51"><i><font color="Green">E32</font></i></a>, </i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E22:51"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4">REALSET1:def 4</a></i>;<br/>
<a NAME="E23:51"/><i><font color="Green">E62</font></i>: 
(  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">A2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E13:51"><i><font color="Green">E42</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E24:51"/><i><font color="Green">E63</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Maroon">o2</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">o2</font>,<font color="Maroon">o</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">o1</font>,<font color="Maroon">o</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">o</font>,<font color="Maroon">o3</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">o</font>,<font color="Maroon">o</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">o2</font>,<font color="Maroon">o3</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_3.html#T2" target="_self">Th2</a>, <a class="ref" href="altcat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_3.html#T4" target="_self">Th4</a>, <a class="ref" href="altcat_3.html#D5" target="_self">Def5</a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<b>thus </b><font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">M1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A2</font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="altcat_3.html#E8:51"><i><font color="Green">E31</font></i></a>, <a class="ref" href="altcat_3.html#T5" target="_self">Th5</a>, <a class="txt" href="altcat_3.html#E6:51"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A2</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B1</font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A1</font>
<b>by </b><i><a class="ref" href="altcat_3.html#D6" target="_self">Def6</a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <span class="p2">(<span class="default"><a href="altcat_1.html#K8">idm</a> <font color="Maroon">o</font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="altcat_3.html#T2" target="_self">Th2</a>, <a class="ref" href="altcat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_3.html#D5" target="_self">Def5</a>, <a class="ref" href="altcat_3.html#T7" target="_self">Th7</a>, <a class="txt" href="altcat_3.html#E12:51"><i><font color="Green">E41</font></i></a>, <a class="txt" href="altcat_3.html#E15:51"><i><font color="Green">E43</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="altcat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>
;<br/>


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