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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</a>  implies for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">a</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">b</font> )</span><br/>

<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="yellow18.html#R2" title="YELLOW18:pred.2">are_opposite</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#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Olive" title="b1">a</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">b</font></span><br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</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">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</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">b</font></span><br/><b>let </b><font color="Maroon" title="c4">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> implies  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</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">b</font> )</span><br/>



<b>assume </b><a NAME="E2:28"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</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">a</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">b</font></span><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">b</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow18.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</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">a</font>,<font color="Maroon" title="c6">c</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">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">c</font> holds  <font color="Olive" title="b1">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E13">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">a</font>,<font color="Maroon" title="c6">c</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">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">c</font> holds  <font color="Olive" title="b1">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c7">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">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="c7">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font></span><br/><b>reconsider </b><font color="Maroon" title="c8">d</font> = <font color="Maroon" title="c6">c</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="yellow18.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/><a NAME="E3:28_1"/><i><font color="Green" title="E14">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="c3">a</font>,<font color="Maroon" title="c6">c</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow18.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c4">b</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow18.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>;<br/><b>thus </b><font color="Maroon" title="c7">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">i</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">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:28"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="yellow18.html#E2:28"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="yellow18.html#E1:28_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:28_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="yellow18.html#E3:28_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</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="E5:28"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</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">b</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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