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

<span class="kw">let </span><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> holds <br/>( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font> implies ( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:53"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="wellord1.html#R3" title="WELLORD1:pred.3">is_isomorphism_of</a> <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) )</span><br/>

<a NAME="E2:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/>
<a NAME="E3:53"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/>
<a NAME="E4:53"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/>
<a NAME="E5:53"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A5</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" title="FUNCT_1:th.33">FUNCT_1:33</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:53"/>
( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_1"/><span class="lab"><font color="Green" title="E53">A6</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><a NAME="E1:53_1_1"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E2:53_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E3:53_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span></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="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="txt" href="wellord1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><a NAME="E4:53_1_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><span class="kw">hence </span><a NAME="E5:53_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:53_1_1"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53_1"/>
<font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="wellord1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:53"/>
( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_2"/><span class="lab"><font color="Green" title="E53">A9</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">S</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53_2_1"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:53_2_1"/><span class="lab"><font color="Green" title="E55">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">S</font></span><br/><a NAME="E3:53_2_1"/><span class="lab"><font color="Green" title="E56">A12</font></span>: 
<font color="Maroon" title="c6">c</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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_2_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E4:53_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A13</font></span>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><a NAME="E5:53_2_1"/>
<font color="Maroon" title="c5">b</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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E6:53_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A14</font></span>: 
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</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> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E7:53_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E3:53_2_1"><span class="lab"><font color="Green" title="E56">A12</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E8:53_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span></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="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E2:53_2_1"><span class="lab"><font color="Green" title="E55">A11</font></span></a>, <a class="txt" href="#E4:53_2_1"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="txt" href="#E6:53_2_1"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/><a NAME="E9:53_2_1"/><span class="lab"><font color="Green" title="E60">A16</font></span>: 
<font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E10:53_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A17</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><a NAME="E11:53_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E9:53_2_1"><span class="lab"><font color="Green" title="E60">A16</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E12:53_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span></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="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E1:53_2_1"><span class="lab"><font color="Green" title="E54">A10</font></span></a>, <a class="txt" href="#E10:53_2_1"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E6:53_2_1"><span class="lab"><font color="Green" title="E58">A14</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/><a NAME="E13:53_2_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font></span>)</span></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="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_2"><span class="lab"><font color="Green" title="E53">A9</font></span></a>, <a class="txt" href="#E8:53_2_1"><span class="lab"><font color="Green" title="E59">A15</font></span></a>, <a class="txt" href="wellord1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:53_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E10:53_2_1"><span class="lab"><font color="Green" title="E61">A17</font></span></a>, <a class="txt" href="#E4:53_2_1"><span class="lab"><font color="Green" title="E57">A13</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53_2"/>
<font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="wellord1.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E8:53"/>
( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  ) &amp; ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_3"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">b</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> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font> &amp;  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53_3_1"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
( <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c5">b</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> <font color="Maroon" title="c2">S</font> )
 <span class="kw">and </span><br/><a NAME="E2:53_3_1"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font> )</span><br/><a NAME="E3:53_3_1"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E1:53_3_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><a NAME="E4:53_3_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c1">R</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</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> <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E1:53_3_1"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E5:53_3_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span></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="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span></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="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_3"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="txt" href="#E2:53_3_1"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E3:53_3_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="wellord1.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:53_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:53_3_1"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53_3"/>
<font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="wellord1.html#E4"><span class="lab"><font color="Green" title="E4">Lm4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E9:53"/>
( <font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  implies <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:53_4"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font> implies <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53_4_1"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E2:53_4_1"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font></span><br/><a NAME="E3:53_4_1"/><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<font color="Maroon" title="c4">a</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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_4_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E4:53_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><a NAME="E5:53_4_1"/><span class="lab"><font color="Green" title="E58">A27</font></span>: 
<font color="Maroon" title="c5">b</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> <font color="Maroon" title="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_4_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="relat_1.html#T15" title="RELAT_1:th.15">RELAT_1:15</a></span>;<br/><a NAME="E6:53_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A28</font></span>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/><a NAME="E7:53_4_1"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E5:53_4_1"><span class="lab"><font color="Green" title="E58">A27</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E8:53_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E3:53_4_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/><a NAME="E9:53_4_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span></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="c1">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">b</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font></span>)</span></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="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E1:53_4_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E2:53_4_1"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="txt" href="#E4:53_4_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E6:53_4_1"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="#E7:53_4_1"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E10:53_4_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53_4"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E4:53_4_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E6:53_4_1"><span class="lab"><font color="Green" title="E59">A28</font></span></a>, <a class="txt" href="wellord1.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53_4"/>
<font color="Maroon" title="c2">S</font> is  <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="wellord1.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E10:53"/><span class="lab"><font color="Green" title="E53">A30</font></span>: 
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b2">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">Y</font> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord1.html#D2" target="_self" title="WELLORD1:def.2">WELLORD1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">Z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord1.html#D2" target="_self" title="WELLORD1:def.2">WELLORD1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E11:53"/><span class="lab"><font color="Green" title="E54">A31</font></span>: 
<font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c2">S</font>
 <span class="kw">and </span><br/><a NAME="E12:53"/><span class="lab"><font color="Green" title="E55">A32</font></span>: 
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Olive" title="b1">a</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<a NAME="E13:53"/><span class="lab"><font color="Green" title="E56">A33</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E15:53"/><span class="lab"><font color="Green" title="E57">A34</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">and </span><br/><a NAME="E16:53"/><span class="lab"><font color="Green" title="E58">A35</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">x</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E10:53"><span class="lab"><font color="Green" title="E53">A30</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E12:53"><span class="lab"><font color="Green" title="E55">A32</font></span></a>, <a class="ref" href="relat_1.html#T139" title="RELAT_1:th.139">RELAT_1:139</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<span class="kw">thus </span><a NAME="E17:53"/>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:53"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font></span><br/>

<span class="kw">assume </span><a NAME="E18:53"/>
 not <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E20:53"/><span class="lab"><font color="Green" title="E59">A36</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E21:53"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E22:53"/><span class="lab"><font color="Green" title="E61">A38</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E5:53"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E23:53"/><span class="lab"><font color="Green" title="E62">A39</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font></span>)</span></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">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:53"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="ref" href="wellord1.html#T1" target="_self" title="WELLORD1:th.1">Th1</a></span>;<br/>
<a NAME="E24:53"/><span class="lab"><font color="Green" title="E63">A40</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:53"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E4:53"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E11:53"><span class="lab"><font color="Green" title="E54">A31</font></span></a>, <a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E25:53"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <font color="Maroon" title="c4">Z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:53"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="txt" href="#E22:53"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="funct_1.html#D7" title="FUNCT_1:def.7">FUNCT_1:def 7</a></span>;<br/>
<a NAME="E26:53"/><span class="kw">then </span>
 not <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K1" title="WELLORD1:func.1">-Seg</a> <font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:53"><span class="lab"><font color="Green" title="E58">A35</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E27:53"/><span class="kw">then </span>
(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span>,<font color="Maroon" title="c5">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="c1">R</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T1" target="_self" title="WELLORD1:th.1">Th1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:53"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E13:53"><span class="lab"><font color="Green" title="E56">A33</font></span></a>, <a class="txt" href="#E15:53"><span class="lab"><font color="Green" title="E57">A34</font></span></a>, <a class="txt" href="#E20:53"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="txt" href="#E24:53"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="txt" href="#E22:53"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="txt" href="#E23:53"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="ref" href="wellord1.html#T1" target="_self" title="WELLORD1:th.1">Th1</a>, <a class="ref" href="wellord1.html#D7" target="_self" title="WELLORD1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
