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

<span class="kw">consider </span><font color="Maroon" title="c1">R</font> being   <a href="relset_1.html#NM2" title="RELSET_1:NM.2">Relation</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E2:202_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="c1">R</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="relset_1.html#S2" title="RELSET_1:sch.2">RELSET_1:sch 2</a>();<br/></span>
<a NAME="E3:202_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive">b<sub>1</sub></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  not <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="Olive">b<sub>2</sub></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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive">b<sub>2</sub></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> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c4">z</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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:202_1_1"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:202_1_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E3:202_1_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">x9</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">y9</font> = <font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c7">z9</font> = <font color="Maroon" title="c4">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202_1_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E2:202_1_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E3:202_1_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:202_1_1"/><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E6:202_1_1"/><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c2">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E7:202_1_1"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c6">y9</font>,<font color="Maroon" title="c7">z9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E6:202_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a></span>;<br/>
<a NAME="E8:202_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c6">y9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E5:202_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a></span>;<br/>
<a NAME="E9:202_1_1"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x9</font>,<font color="Maroon" title="c7">z9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:202"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E7:202_1_1"><span class="lab"><font color="Green" title="E61">A12</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:202_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:202_1"/>
<font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:202_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">x9</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E3:202_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x9</font>,<font color="Maroon" title="c3">x9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<a NAME="E4:202_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:202_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:202_1"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E6:202_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A13</font></span>: 
<font color="Maroon">F<sub>1</sub></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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<a NAME="E7:202_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">R</font></span>)</span>
 
;<br/>
<a NAME="E8:202_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A14</font></span>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:202_1"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E9:202_1"/><span class="lab"><font color="Green" title="E58">A15</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:202_1_3"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">x9</font> = <font color="Maroon" title="c2">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E3:202_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x9</font>,<font color="Maroon" title="c3">x9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202"><span class="lab"><font color="Green" title="E50">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:202_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:202_1"/><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon">F<sub>1</sub></font>() <a href="orders_1.html#R5" title="ORDERS_1:pred.5">has_lower_Zorn_property_wrt</a> <font color="Maroon" title="c1">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_1.html#D10" target="_self" title="ORDERS_1:def.10">ORDERS_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a>  implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="c1">R</font> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:202_1_4"/><span class="lab"><font color="Green" title="E60">A17</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:202_1_4"/><span class="lab"><font color="Green" title="E61">A18</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="c1">R</font> ) )</span><br/>

<a NAME="E3:202_1_4"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:202_1_4_1"/><span class="lab"><font color="Green" title="E62">A19</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E2:202_1_4_1"/><span class="lab"><font color="Green" title="E63">A20</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>] )</span><br/>

<a NAME="E3:202_1_4_1"/><span class="lab"><font color="Green" title="E64">A21</font></span>: 
( <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> is  <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>  &amp; <font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</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="#E2:202_1_4"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="ref" href="orders_1.html#D5" target="_self" title="ORDERS_1:def.5">Def5</a></span>;<br/>
<a NAME="E4:202_1_4_1"/>
<font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> or <font color="Maroon" title="c5">x</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"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:202_1_4_1_1"/><span class="lab"><font color="Green" title="E65">A22</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</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"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E2:202_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A23</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c2">Y</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E3:202_1_4_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:202_1"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E1:202_1_4"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E1:202_1_4_1_1"><span class="lab"><font color="Green" title="E65">A22</font></span></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></span>;<br/>
<a NAME="E4:202_1_4_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<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> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1_4_1_1"><span class="lab"><font color="Green" title="E66">A23</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:202_1_4_1_1"/>
<font color="Maroon" title="c5">x</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"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E65">A24</font></span>: <font color="Maroon" title="c2">Y</font> = 
 <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"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:202_1_4_1"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="orders_1.html#E122"><span class="lab"><font color="Green" title="E16">Lm5</font></span></a></span>
<br/>.= 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T28" title="WELLORD1:th.28">WELLORD1:28</a></span>
;<br/>
<a NAME="E6:202_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A25</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:202_1_4_1"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="ref" href="relat_2.html#D14" title="RELAT_2:def.14">RELAT_2:def 14</a></span>;<br/>
<a NAME="E7:202_1_4_1"/><span class="lab"><font color="Green" title="E67">A26</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:202_1_4_1"><span class="lab"><font color="Green" title="E64">A21</font></span></a>, <a class="txt" href="#E5:202_1_4_1"><span class="lab"><font color="Green" title="E65">A24</font></span></a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></span>;<br/>
<a NAME="E8:202_1_4_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font> )
 
;<br/>
<a NAME="E9:202_1_4_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c1">R</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">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> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202_1_4_1"><span class="lab"><font color="Green" title="E62">A19</font></span></a>, <a class="txt" href="#E2:202_1_4_1"><span class="lab"><font color="Green" title="E63">A20</font></span></a>, <a class="txt" href="#E6:202_1_4_1"><span class="lab"><font color="Green" title="E66">A25</font></span></a>, <a class="txt" href="#E7:202_1_4_1"><span class="lab"><font color="Green" title="E67">A26</font></span></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a>, <a class="ref" href="relat_2.html#D6" title="RELAT_2:def.6">RELAT_2:def 6</a></span>;<br/>
<a NAME="E10:202_1_4_1"/><span class="kw">then </span>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">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="c1">R</font> or <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:202_1_4_1"/>
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c3">x</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<span class="kw"> such that </span><br/><a NAME="E5:202_1_4"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:202"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E1:202_1_4"><span class="lab"><font color="Green" title="E60">A17</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c3">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">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="c1">R</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:202_1_4"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Olive" title="b1">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="c1">R</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">x</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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> )</span><br/>

<span class="kw">assume </span><a NAME="E7:202_1_4"/><span class="lab"><font color="Green" title="E63">A28</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c3">y</font>,<font color="Maroon" title="c4">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></span><br/>

<a NAME="E8:202_1_4"/><span class="kw">then </span>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">x</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202_1_4"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E5:202_1_4"><span class="lab"><font color="Green" title="E62">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:202_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E1:202_1_4"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E7:202_1_4"><span class="lab"><font color="Green" title="E63">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:202_1"/>
<font color="Maroon" title="c1">R</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="202_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="relat_2.html#D4" title="RELAT_2:def.4">RELAT_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Olive">b<sub>1</sub></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  not <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="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:202_1_5"/><span class="lab"><font color="Green" title="E60">A29</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">and </span><br/><a NAME="E2:202_1_5"/><span class="lab"><font color="Green" title="E61">A30</font></span>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y9</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:202_1_5"><span class="lab"><font color="Green" title="E60">A29</font></span></a>, <a class="txt" href="#E2:202_1_5"><span class="lab"><font color="Green" title="E61">A30</font></span></a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:202_1_5"/><span class="lab"><font color="Green" title="E62">A31</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">and </span><br/><a NAME="E5:202_1_5"/><span class="lab"><font color="Green" title="E63">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E6:202_1_5"/><span class="lab"><font color="Green" title="E64">A33</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">y9</font>,<font color="Maroon" title="c4">x9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E5:202_1_5"><span class="lab"><font color="Green" title="E63">A32</font></span></a></span>;<br/>
<a NAME="E7:202_1_5"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c4">x9</font>,<font color="Maroon" title="c5">y9</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E4:202_1_5"><span class="lab"><font color="Green" title="E62">A31</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:202_1_5"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E6:202_1_5"><span class="lab"><font color="Green" title="E64">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:202_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">R</font> <a href="orders_1.html#R2" title="ORDERS_1:pred.2">partially_orders</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:202_1"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E3:202_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="ref" href="orders_1.html#D7" target="_self" title="ORDERS_1:def.7">Def7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:202_1"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="orders_1.html#R7" title="ORDERS_1:pred.7">is_minimal_in</a> <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:202_1"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E10:202_1"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="orders_1.html#T174" target="_self" title="ORDERS_1:th.174">Th174</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c2">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</font>] ) )</span><br/>

<span class="kw">thus </span><a NAME="E15:202_1"/>
<font color="Maroon" title="c2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:202_1"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E14:202_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="orders_1.html#D12" target="_self" title="ORDERS_1:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> holds <br/> not <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c2">x</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font> implies  not <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E16:202_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>]</span><br/>

<a NAME="E17:202_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A35</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:202_1"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E14:202_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="orders_1.html#D12" target="_self" title="ORDERS_1:def.12">Def12</a></span>;<br/>
<a NAME="E18:202_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:202_1"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="txt" href="#E14:202_1"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="orders_1.html#D12" target="_self" title="ORDERS_1:def.12">Def12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:202_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:202_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E17:202_1"><span class="lab"><font color="Green" title="E61">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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