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

<b>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">cR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>thus </b><a NAME="E1:46_1_1"/>
( <font color="Maroon">R</font> is <a href="neckla_3.html#V1" target="_self">path-connected</a> implies for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/>the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">x</font>,<font color="Olive">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:46_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> is <a href="neckla_3.html#V1" target="_self">path-connected</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:46_1_1_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_1_1"/>
( the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> or the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> )
 </span><b>by </b><i>, , </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_1_1_1"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_1_1_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_1_1_2"/>
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>assume </b><a NAME="E2:46_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> holds <br/>the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 ;<br/>

<a NAME="E3:46_1_1"/><b>then </b>
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; not the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">x</font>,<font color="Olive">y</font> holds <br/>the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> <a href="rewrite1.html#R1">reduces</a> <font color="Olive">y</font>,<font color="Olive">x</font>
 
;<br/>
<b>hence </b><a NAME="E4:46_1_1"/>
<font color="Maroon">R</font> is <a href="neckla_3.html#V1" target="_self">path-connected</a>
 <b>by </b><i/>;<br/>


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