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

<a NAME="E1:19_1_1"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="orders_1.html#T100" title="ORDERS_1:th.100">ORDERS_1:100</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> &amp; not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font> )</span><br/><b>assume </b><b>that </b><br/><a NAME="E1:19_1_1_1"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E2:19_1_1_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E3:19_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 <b>and </b><br/><a NAME="E4:19_1_1_1"/><i><font color="Green" title="E14">A4</font></i>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font></span><br/><b>reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">y</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E2:19_1_1_1"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/><a NAME="E6:19_1_1_1"/>
not <font color="Maroon" title="c4">p</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="txt" href="#E4:19_1_1_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</a></i>;<br/><a NAME="E7:19_1_1_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="polynom3.html#NR2" title="POLYNOM3:NR.2">&gt;</a> <font color="Maroon" title="c5">q</font>
 <b>by </b><i><a class="ref" href="polynom3.html#T7" target="_self" title="POLYNOM3:th.7">Th7</a></i>;<br/><a NAME="E8:19_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="polynom3.html#R2" title="POLYNOM3:pred.2">&lt;=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="polynom3.html#D2" target="_self" title="POLYNOM3:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E9:19_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="polynom3.html#D3" target="_self" title="POLYNOM3:def.3">Def3</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>
<a NAME="E3:19_1_1"/><b>then </b>
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/>
<a NAME="E4:19_1_1"/><b>then </b>
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 
<b>by </b><i><a class="txt" href="#E1:19_1_1"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1"/>
 <a href="polynom3.html#K5" title="POLYNOM3:func.5">TuplesOrder</a> <font color="Maroon" title="c1">n</font> is <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>
 <b>by </b><i><a class="ref" href="orders_1.html#D5" title="ORDERS_1:def.5">ORDERS_1:def 5</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>
