<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/>

<b>assume </b><a NAME="E1:31"/><i><font color="Green" title="E14">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> 
 ;<br/>

<b>assume </b><a NAME="E2:31"/>
<font color="Maroon" title="c1">A</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">A</font> = <font color="Maroon" title="c1">A</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">A</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> $3,$2,$1</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> $4,$5;<br/>
<div><i><font color="Green" title="E15">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <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="c2">B</font>;<br/><b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E16">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="c4">a</font>,<font color="Maroon" title="c5">b</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</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>  )
 ;<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="c4">a</font>,<font color="Maroon" title="c5">b</font>;<br/><b>let </b><font color="Maroon" title="c8">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c11">c'</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="c3">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self" title="YELLOW18:th.6">Th6</a></i>;<br/><a NAME="E3:31_1"/><i><font color="Green" title="E17">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">a</font>,<font color="Maroon" title="c5">b</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="c10">b'</font>,<font color="Maroon" title="c9">a'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c5">b</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="c11">c'</font>,<font color="Maroon" title="c10">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#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">f'</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="c10">b'</font>,<font color="Maroon" title="c9">a'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">g'</font> = <font color="Maroon" title="c8">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c10">b'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>thus </b><font color="Maroon" title="c8">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">f</font> = 
<font color="Maroon" title="c12">f'</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c13">g'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="yellow18.html#E1:31_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self" title="YELLOW18:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:31_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="yellow18.html#E3:31_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E16">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self" title="YELLOW18:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">a'</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font>;<br/><a NAME="E3:31_2"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">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="c5">a'</font>,<font color="Maroon" title="c5">a'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>hence </b><a NAME="E4:31_2"/>
<font color="Maroon" title="c7">f</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">a</font>,<font color="Maroon" title="c4">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="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c9">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>reconsider </b><font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self" title="YELLOW18:th.6">Th6</a></i>;<br/><b>assume </b><a NAME="E6:31_2"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c9">g</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">a</font>,<font color="Maroon" title="c8">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><a NAME="E7:31_2"/><b>then </b><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c9">g</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="c10">b'</font>,<font color="Maroon" title="c5">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="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c5">a'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="yellow18.html#E6:31_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c9">g</font>) = 
<span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">a'</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c11">g'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E7:31_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E7:31_2"><i><font color="Green" title="E18">A7</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>
<div><i><font color="Green" title="E17">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>;<br/><b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self" title="YELLOW18:th.6">Th6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> =  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">a'</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f</font>;<br/><a NAME="E3:31_3"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">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="c5">a'</font>,<font color="Maroon" title="c5">a'</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><b>hence </b><a NAME="E4:31_3"/>
<font color="Maroon" title="c7">f</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">a</font>,<font color="Maroon" title="c4">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="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c9">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>reconsider </b><font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c8">b</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self" title="YELLOW18:th.6">Th6</a></i>;<br/><b>assume </b><a NAME="E6:31_3"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c9">g</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="c8">b</font>,<font color="Maroon" title="c4">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/><a NAME="E7:31_3"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c9">g</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="c5">a'</font>,<font color="Maroon" title="c10">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#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">g'</font> = <font color="Maroon" title="c9">g</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c10">b'</font> <b>by </b><i><a class="txt" href="yellow18.html#E1:31"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="yellow18.html#E6:31_3"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self" title="YELLOW18:th.7">Th7</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c9">g</font>,<font color="Maroon" title="c7">f</font>) = 
<font color="Maroon" title="c11">g'</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="c5">a'</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E7:31_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="altcat_1.html#D10" title="ALTCAT_1:def.10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon" title="c9">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E7:31_3"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E7:31"/>
<font color="Maroon" title="c2">B</font> is <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S3" target="_self" title="YELLOW18:sch.3">YELLOW18:sch 3</a>(<a class="txt" href="yellow18.html#E4:31"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="yellow18.html#E5:31"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="yellow18.html#E6:31"><i><font color="Green" title="E17">A8</font></i></a>);</i><br/>


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