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<b>let </b><font color="Maroon">C</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a> <a href="altcat_1.html#V11">associative</a>  <a href="altcat_1.html#L2">AltCatStr</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>assume </b><a NAME="E1:19"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o1</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">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>  )
 ;<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">o2</font>;<br/><b>let </b><font color="Maroon">B</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><a NAME="E2:19"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">A</font> is <a href="altcat_3.html#V5" target="_self">epi</a> &amp; <font color="Maroon">B</font> is <a href="altcat_3.html#V5" target="_self">epi</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#D8" target="_self">ALTCAT_3:def 8</a><br/>

<b>assume </b><a NAME="E3:19"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">o3</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> 
 ;<br/>

<b>let </b><font color="Maroon">M1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o</font>, <font color="Maroon">M2</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o3</font>,<font color="Maroon">o</font>;<br/>

<b>assume </b><a NAME="E4:19"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<a NAME="E5:19"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font></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#D3" target="_self">Def3</a>, , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font></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#D3" target="_self">Def3</a>, , <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>;<br/>
<a NAME="E7:19"/>
<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> 
 
<b>by </b><i><a class="ref" href="altcat_3.html#D3" target="_self">Def3</a>, , <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/>
<a NAME="E8:19"/><b>then </b>
<font color="Maroon">M1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="altcat_3.html#T1" target="_self">Th1</a>, <a class="txt" href="altcat_3.html#E1:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="altcat_3.html#E2:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="altcat_3.html#D4" target="_self">Def4</a>, </i>;<br/>
<b>hence </b><a NAME="E9:19"/>
<font color="Maroon">M1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="altcat_3.html#T1" target="_self">Th1</a>, , </i>;<br/>


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