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<b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><a NAME="E1:48"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green">E36</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">a</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#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/>

<b>assume </b><a NAME="E3:48"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E4:48"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b'</font>,<font color="Maroon">a'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">a'</font>,<font color="Maroon">b'</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">B</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A</font>;<br/><b>assume </b><a NAME="E2:48_1"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</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">b</font>,<font color="Maroon">a</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#NM1">object</a> of <font color="Maroon">B</font>, <font color="Maroon">b'</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">B</font>;<br/><b>assume </b><a NAME="E3:48_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/><b>let </b><font color="Maroon">f'</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">a'</font>;<br/><b>assume </b><a NAME="E4:48_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 ;<br/><b>assume </b><a NAME="E5:48_1"/>
<font color="Maroon">f</font> is <a href="altcat_3.html#V3">iso</a>
 ;<br/><a NAME="E6:48_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">f</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> is <a href="altcat_3.html#V1">retraction</a> &amp; <font color="Maroon">f</font> is <a href="altcat_3.html#V2">coretraction</a> )
 <b>by </b><i><a class="ref" href="altcat_3.html#T5">ALTCAT_3:5</a>, <a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>;<br/><a NAME="E7:48_1"/><b>then </b>
( <font color="Maroon">f'</font> <a href="altcat_3.html#K1">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="altcat_3.html#K1">"</a>  &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a'</font> &amp;  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E8:48_1"/><b>then </b>
( <font color="Maroon">f'</font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">a'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="altcat_3.html#K1">"</a> </span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">f'</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K8">idm</a> <font color="Maroon">b'</font> )
 <b>by </b><i>, , , , , </i>;<br/><b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon">f'</font> is <a href="altcat_3.html#V3">iso</a>
 <b>by </b><i><a class="ref" href="altcat_3.html#D5">ALTCAT_3:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:48"/>
for <font color="Olive">f</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font><br/> for <font color="Olive">f'</font> being  <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">b'</font>,<font color="Maroon">a'</font>  st <font color="Olive">f'</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/>( <font color="Olive">f</font> is <a href="altcat_3.html#V3">iso</a> iff <font color="Olive">f'</font> is <a href="altcat_3.html#V3">iso</a> )
 <b>by </b><i>, , , </i>;<br/>


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