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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<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>set </b><font color="Maroon">IR</font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>;<br/>
<b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">R</font> is <a href="waybel_0.html#V16">connected</a>
 ;<br/>

<a NAME="E2:42_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> &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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<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="E1:42_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/>

<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E3:42_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font> or <font color="Maroon">y</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> )
 </span><b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="rewrite1.html#M1">RedSequence</a> of the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T8">REWRITE1:8</a></i>;<br/><a NAME="E3:42_1_1_1_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:42_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> 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> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> is    <a href="rewrite1.html#M1">RedSequence</a> of the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="rewrite1.html#T8">REWRITE1:8</a></i>;<br/><a NAME="E3:42_1_1_1_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="finseq_1.html#K10">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>hence </b><a NAME="E4:42_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> 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> )
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D3">REWRITE1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:42_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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