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<b>let </b><font color="Maroon" title="c1">A</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>;<br/><b>let </b><font color="Maroon" title="c2">B</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c3">BB</font> = <font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  <b>by </b><i><a class="ref" href="yellow20.html#T48" target="_self" title="YELLOW20:th.48">Th48</a></i>;<br/>
<a NAME="E2:63"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c3">BB</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> ,<font color="Maroon" title="c1">A</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  )
 
<b>by </b><i><a class="ref" href="yellow18.html#D4" title="YELLOW18:def.4">YELLOW18:def 4</a></i>;<br/>
<a NAME="E3:63"/>
<font color="Maroon" title="c3">BB</font> is <a href="altcat_2.html#V3" title="ALTCAT_2:attr.3">id-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1_1"/>
(  not <font color="Maroon" title="c3">BB</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c3">BB</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_1"><b>case </b></a><a NAME="E1:63_1_1_1"/>
not <font color="Maroon" title="c3">BB</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c4">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">BB</font>;<br/><b>let </b><font color="Maroon" title="c5">o'</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> </span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c4">o</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font> <b>by </b><i><a class="txt" href="yellow20.html#E2:63"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#D3" title="YELLOW18:def.3">YELLOW18:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c5">o'</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> <b>by </b><i><a class="txt" href="yellow20.html#E2:63"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#D3" title="YELLOW18:def.3">YELLOW18:def 3</a></i>;<br/><b>assume </b><a NAME="E4:63_1_1_1"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">o'</font>
 ;<br/><a NAME="E5:63_1_1_1"/><b>then </b>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c7">a'</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="c6">a</font>,<font color="Maroon" title="c6">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="ref" href="altcat_2.html#D14" title="ALTCAT_2:def.14">ALTCAT_2:def 14</a></i>;<br/><a NAME="E6:63_1_1_1"/><b>then </b>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">o'</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="c6">a</font>,<font color="Maroon" title="c6">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:63"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#T10" title="YELLOW18:th.10">YELLOW18:10</a></i>;<br/><b>hence </b><a NAME="E7:63_1_1_1"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">o'</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="c4">o</font>,<font color="Maroon" title="c4">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow20.html#E2:63"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" title="YELLOW18:th.7">YELLOW18:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1_2"><b>case </b></a><a NAME="E1:63_1_1_2"/>
<font color="Maroon" title="c3">BB</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:63_1_1_2"/>
verum
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:63"/>
<font color="Maroon" title="c2">B</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a>  is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font> <a href="yellow18.html#K1" title="YELLOW18:func.1">opp</a> 
 ;<br/>


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