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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">D</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">D</font><br/> for <font color="Olive" title="b3">o'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">o'</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b3">o'</font></span><br/><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#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font><br/> for <font color="Olive" title="b2">o'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">o'</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b2">o'</font></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">o'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o'</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">o'</font></span><br/><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="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o'</font> implies  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o'</font> )</span><br/>







<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o'</font></span><br/>

<a NAME="E2:75"/><b>then </b><i><font color="Green" title="E36">A2</font></i>: 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">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="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="ref" href="altcat_2.html#D14" target="_self" title="ALTCAT_2:def.14">Def14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">m</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o'</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="#E1:75"><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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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">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>  implies for <font color="Olive" title="b1">a</font> being  <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">p</font> holds  <font color="Olive" title="b1">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> )</span><br/><b>assume </b><a NAME="E1:75_1"/><i><font color="Green" title="E37">A3</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">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <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">p</font> holds  <font color="Olive" title="b1">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font></span><br/><b>let </b><font color="Maroon" title="c7">a</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">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font></span><br/><b>reconsider </b><font color="Maroon" title="c8">p'</font> = <font color="Maroon" title="c6">p</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="E3:75_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="c4">o'</font>,<font color="Maroon" title="c8">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="#E1:75"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E1:75_1"><i><font color="Green" title="E37">A3</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="c9">n</font> = <font color="Maroon" title="c7">a</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o'</font>,<font color="Maroon" title="c8">p'</font> <b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E1:75_1"><i><font color="Green" title="E37">A3</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="c7">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">m</font> = 
<font color="Maroon" title="c9">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">o'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:75"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="#E2:75"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="#E1:75_1"><i><font color="Green" title="E37">A3</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">a</font>
<b>by </b><i><a class="txt" href="#E3:75_1"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:75"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">o'</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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