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<b>let </b><font color="Maroon" title="c2">D</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="altcat_2.html#M1" title="ALTCAT_2:mode.1">SubCatStr</a> of <font color="Maroon" title="c1">C</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:73_1_1"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">D</font> is <a href="altcat_2.html#V3" title="ALTCAT_2:attr.3">id-inheriting</a>
 <b>and </b><br/><a NAME="E2:73_1_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c2">D</font> is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_2.html#D9" target="_self" title="ALTCAT_2:def.9">ALTCAT_2:def 9</a><br/>

<b>reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">o</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" target="_self" title="ALTCAT_2:th.30">Th30</a></i>;<br/>
<a NAME="E4:73_1_1"/><i><font color="Green" title="E37">A3</font></i>: 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">p</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="c3">o</font>,<font color="Maroon" title="c3">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="altcat_2.html#E1:73_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="altcat_2.html#D14" target="_self" title="ALTCAT_2:def.14">Def14</a></i>;<br/>
<b>hence </b><a NAME="E5:73_1_1"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c3">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">i</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">p</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c3">o</font> <b>by </b><i><a class="txt" href="altcat_2.html#E1:73_1_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="altcat_2.html#D14" target="_self" title="ALTCAT_2:def.14">Def14</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">i</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">o'</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c7">m'</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c6">o'</font>,<font color="Maroon" title="c3">o</font>;<br/><b>let </b><font color="Maroon" title="c8">m''</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">o'</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:73_1_1_1"/><i><font color="Green" title="E38">A4</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o'</font>,<font color="Maroon" title="c3">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c9">p'</font> = <font color="Maroon" title="c6">o'</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" target="_self" title="ALTCAT_2:th.30">Th30</a></i>;<br/><a NAME="E3:73_1_1_1"/><i><font color="Green" title="E39">A5</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c4">p</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="altcat_2.html#E1:73_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="altcat_2.html#T32" target="_self" title="ALTCAT_2:th.32">Th32</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">n'</font> = <font color="Maroon" title="c7">m'</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">p'</font>,<font color="Maroon" title="c4">p</font> <b>by </b><i><a class="txt" href="altcat_2.html#E1:73_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="altcat_2.html#T34" target="_self" title="ALTCAT_2:th.34">Th34</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">i</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">m'</font> = 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c10">n'</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E2:73_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="altcat_2.html#E4:73_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="altcat_2.html#E1:73_1_1_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c7">m'</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E3:73_1_1_1"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></i>
;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:73_1_1"/><i><font color="Green" title="E38">A6</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o</font>,<font color="Maroon" title="c6">o'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">p'</font> = <font color="Maroon" title="c6">o'</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="altcat_2.html#T30" target="_self" title="ALTCAT_2:th.30">Th30</a></i>;<br/>
<a NAME="E10:73_1_1"/><i><font color="Green" title="E39">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">p'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_2.html#E8:73_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="altcat_2.html#T32" target="_self" title="ALTCAT_2:th.32">Th32</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">n''</font> = <font color="Maroon" title="c8">m''</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c9">p'</font> <b>by </b><i><a class="txt" href="altcat_2.html#E8:73_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="altcat_2.html#T34" target="_self" title="ALTCAT_2:th.34">Th34</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c8">m''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c10">n''</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">p</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_2.html#E2:73_1_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="altcat_2.html#E4:73_1_1"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="altcat_2.html#E8:73_1_1"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="altcat_2.html#T33" target="_self" title="ALTCAT_2:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c8">m''</font>
<b>by </b><i><a class="txt" href="altcat_2.html#E10:73_1_1"><i><font color="Green" title="E39">A7</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
;<br/>


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