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<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">C</font> is <a href="altcat_1.html#V12">with_units</a>
 ;<br/><b>then reconsider </b><font color="Maroon">C'</font> = <font color="Maroon">C</font> as  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</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>;<br/><b>thus </b><a NAME="E3:37_1_1_1"/>
<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> 
 <b>by </b><i>, <a class="ref" href="altcat_1.html#T21">ALTCAT_1:21</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">o</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C'</font> ;<br/><b>reconsider </b><font color="Maroon">i</font> =  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">p</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o</font> ;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>let </b><font color="Maroon">o'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">m'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o'</font>,<font color="Maroon">o</font>;<br/><b>let </b><font color="Maroon">m''</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o'</font>;<br/><b>thus </b><a NAME="E6:37_1_1_1"/>
( <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>  implies <font color="Maroon">i</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m'</font> )
 <b>by </b><i><a class="ref" href="altcat_1.html#T24">ALTCAT_1:24</a></i>;<br/><b>thus </b><a NAME="E7:37_1_1_1"/>
( <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>  implies <font color="Maroon">m''</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m''</font> )
 <b>by </b><i><a class="ref" href="altcat_1.html#D19">ALTCAT_1:def 19</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:37_1_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">o</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> holds <br/> ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o</font>,<font color="Olive">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;  ex <font color="Olive">i</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o</font>,<font color="Olive">o</font> st <br/>for <font color="Olive">o'</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">m'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o'</font>,<font color="Olive">o</font><br/> for <font color="Olive">m''</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o</font>,<font color="Olive">o'</font> holds <br/> ( ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o'</font>,<font color="Olive">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>  implies <font color="Olive">i</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">m'</font> <a href="hidden.html#R1">=</a> <font color="Olive">m'</font> ) &amp; ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">o</font>,<font color="Olive">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>  implies <font color="Olive">m''</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m''</font> ) ) )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D9">ALTCAT_1:def 9</a>,<a class="ref" href="altcat_1.html#D18">ALTCAT_1:def 18</a>
<div class="add"><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ;<br/><b>consider </b><font color="Maroon">m</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o</font><b> such that </b><br/><a NAME="E3:37_1_1_2"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">o'</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">m'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o'</font>,<font color="Maroon">o</font><br/> for <font color="Olive">m''</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Olive">o'</font> holds <br/> ( ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">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>  implies <font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">m'</font> <a href="hidden.html#R1">=</a> <font color="Olive">m'</font> ) &amp; ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Olive">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>  implies <font color="Olive">m''</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m''</font> ) )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">m</font> as    <a href="hidden.html#M1">set</a>  ;<br/><b>take </b><font color="Maroon">e</font> = <font color="Maroon">e</font>;<br/><a NAME="E5:37_1_1_2"/><i><font color="Green">E21</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:37_1_1_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">j</font>,<font color="Maroon">j</font>
 ;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E7:37_1_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/><b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ;<br/><b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o'</font>,<font color="Maroon">o</font> <b>by </b><i><a class="txt" href="altcat_2.html#E5:37_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/><b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e</font>,<font color="Maroon">f</font> = 
<font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m'</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E3:37_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="altcat_2.html#E5:37_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i>, <a class="txt" href="altcat_2.html#E5:37_1_1_2"><i><font color="Green">E21</font></i></a></i>
;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D8">ALTCAT_1:def 8</a><br/>

<b>reconsider </b><font color="Maroon">o</font> = <font color="Maroon">i</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o</font><b> such that </b><br/><a NAME="E6:37_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">o'</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font><br/> for <font color="Olive">m'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Olive">o'</font>,<font color="Maroon">o</font><br/> for <font color="Olive">m''</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Olive">o'</font> holds <br/> ( ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">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>  implies <font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Olive">m'</font> <a href="hidden.html#R1">=</a> <font color="Olive">m'</font> ) &amp; ( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o</font>,<font color="Olive">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>  implies <font color="Olive">m''</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Olive">m''</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">e</font> = <font color="Maroon">m</font>;<br/>

<a NAME="E7:37_1_1"/><i><font color="Green">E30</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:37_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>
 ;<br/>

<b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">f</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E9:37_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">o'</font> = <font color="Maroon">j</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font> ;<br/>
<b>reconsider </b><font color="Maroon">m'</font> = <font color="Maroon">f</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o</font>,<font color="Maroon">o'</font> <b>by </b><i/>;<br/>
<b>thus </b><span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C</font> <a href="altcat_1.html#K4">.</a> <font color="Maroon">i</font>,<font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">f</font>,<font color="Maroon">e</font> = 
<font color="Maroon">m'</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</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>
<b>by </b><i>, </i>
;<br/>


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