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

<b>let </b><font color="Maroon" title="c1">L</font> be    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>;<br/>
<a NAME="E1:30"/><i><font color="Green" title="E14">A1</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:30"/><i><font color="Green" title="E15">A2</font></i>: 
 <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font> <a href="waybel35.html#NR1" title="WAYBEL35:NR.1">is_inductive_wrt</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D9" title="ORDERS_1:def.9">ORDERS_1:def 9</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_1"/><i><font color="Green" title="E16">A3</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E2:30_1"/><i><font color="Green" title="E17">A4</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c5">Y</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">is_linear-order</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1"/>
( <font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/><b>thus </b><a NAME="E2:30_1_1_1"/>
( <font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">C</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="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2"/><i><font color="Green" title="E18">A6</font></i>: 
not <font color="Maroon" title="c5">Y</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c6">Z</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c5">Y</font>;<br/><a NAME="E2:30_1_1_2"/>
<font color="Maroon" title="c6">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_2_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">A</font>
 <b>and </b><br/><a NAME="E4:30_1_1_2_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c8">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:30_1_1_2_1"/>
<font color="Maroon" title="c8">A</font> is    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel35.html#E4:30_1_1_2_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1_1_2_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:30_1_1_2_1"><i><font color="Green" title="E19">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c6">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E4:30_1_1_2"/><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c7">S</font> is    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">WAYBEL35:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_2_2"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A</font>
 <b>and </b><br/><a NAME="E4:30_1_1_2_2"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E5:30_1_1_2_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">B</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:30_1_1_2_2"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">B</font>
 <b>and </b><br/><a NAME="E8:30_1_1_2_2"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E9:30_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c5">Y</font> is <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2:30_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="orders_1.html#D5" title="ORDERS_1:def.5">ORDERS_1:def 5</a></i>;<br/>
<a NAME="E10:30_1_1_2_2"/><b>then </b><i><font color="Green" title="E24">A14</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c5">Y</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p3">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c5">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></i>;<br/>
<a NAME="E11:30_1_1_2_2"/><i><font color="Green" title="E25">A15</font></i>: 
<span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="wellord2.html#T8" title="WELLORD2:th.8">WELLORD2:8</a></i>;<br/>
<a NAME="E12:30_1_1_2_2"/><i><font color="Green" title="E26">A16</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c5">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E13:30_1_1_2_2"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c10">A</font> is    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel35.html#E4:30_1_1_2_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/>
<a NAME="E14:30_1_1_2_2"/><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c11">B</font> is    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel35.html#E8:30_1_1_2_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_1_2_2_1"/>
( <font color="Maroon" title="c10">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">B</font> or <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_1_2_2_1_1"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">B</font>
 ;<br/><div class="add"><a NAME="E2:30_1_1_2_2_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">A</font>,<font color="Maroon" title="c11">B</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="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c5">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">B</font>,<font color="Maroon" title="c10">A</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="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c5">Y</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:30_1_1_2_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="waybel35.html#E8:30_1_1_2_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="waybel35.html#E10:30_1_1_2_2"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="waybel35.html#E11:30_1_1_2_2"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="waybel35.html#E12:30_1_1_2_2"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></i>;<br/><a NAME="E3:30_1_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c10">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">B</font> or <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">A</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E4:30_1_1_2_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="txt" href="waybel35.html#E8:30_1_1_2_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E4:30_1_1_2_2_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E3:30_1_1_2_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="waybel35.html#E7:30_1_1_2_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="waybel35.html#E13:30_1_1_2_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="txt" href="waybel35.html#E14:30_1_1_2_2"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_1_2_2_1_2"/>
<font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_1_1_2_2_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">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> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">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> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E3:30_1_1_2_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="txt" href="waybel35.html#E7:30_1_1_2_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="waybel35.html#E13:30_1_1_2_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:30_1_1_2"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1_1_2_3"/><i><font color="Green" title="E20">A19</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:30_1_1_2_3"/><i><font color="Green" title="E21">A20</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30_1_1_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:30_1_1_2_3"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">y</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel35.html#E3:30_1_1_2_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_1_2_3"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30_1_1_2_3"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="waybel35.html#E3:30_1_1_2_3"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:30_1_1_2"/><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c6">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:30_1_1_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E7:30_1_1_2"/><i><font color="Green" title="E21">A22</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 ;<br/><a NAME="E8:30_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Z</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E9:30_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c6">Z</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="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="waybel35.html#E6:30_1_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="txt" href="waybel35.html#E7:30_1_1_2"><i><font color="Green" title="E21">A22</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<a NAME="E3:30"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_2"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3"><i><font color="Green" title="E3">Lm3</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" title="ORDERS_1:def.7">ORDERS_1:def 7</a><br/>

<b>thus </b><a NAME="E2:30_2"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4"><i><font color="Green" title="E4">Lm4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:30_2"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E5"><i><font color="Green" title="E5">Lm5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">D</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:30"/><i><font color="Green" title="E16">A23</font></i>: 
<font color="Maroon" title="c5">D</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="waybel35.html#E2:30"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="orders_1.html#T173" title="ORDERS_1:th.173">ORDERS_1:173</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">D</font>
;<br/>

<b>thus </b><a NAME="E6:30"/>
<font color="Maroon" title="c5">D</font> <a href="orders_1.html#R6" title="ORDERS_1:pred.6">is_maximal_in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p1">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel35.html#E5:30"><i><font color="Green" title="E16">A23</font></i></a></i>;<br/>

<a NAME="E7:30"/>
<font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel35.html#K2" title="WAYBEL35:func.2">Strict_Chains</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E5:30"><i><font color="Green" title="E16">A23</font></i></a>, <a class="ref" href="orders_1.html#D11" title="ORDERS_1:def.11">ORDERS_1:def 11</a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:30"><i><font color="Green" title="E14">A1</font></i></a>, <a class="ref" href="waybel35.html#D5" target="_self" title="WAYBEL35:def.5">Def5</a></i>;<br/>


</div>
