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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">R</font> is  <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</a>  implies  ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> (<font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">cR</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">IR</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">assume </span><a NAME="E1:51"/>
 not <font color="Maroon" title="c1">R</font> is  <a href="neckla_3.html#V1" title="NECKLA_3:attr.1">path-connected</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">G1</font>, <font color="Olive" title="b2">G2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> (<font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 <span class="kw">and </span><br/><a NAME="E4:51"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">and </span><br/><a NAME="E5:51"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 not  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="neckla_3.html#D2" target="_self" title="NECKLA_3:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E29">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">A1</font> =  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">A2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">A2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">G1</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">G2</font> =  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font>;<br/>
<a NAME="E8:51"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E9:51"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</font></span><br/>

<span class="kw">assume </span><a NAME="E2:51_1"/>
 not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</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/>

<a NAME="E3:51_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp;  not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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>
<a NAME="E10:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A5</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">A2</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="E11:51"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E12:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A7</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/>
<a NAME="E13:51"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">IG1</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">IG2</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E1:51_2"/>
 not  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:51_2"/><span class="kw">then </span>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:51_2"/><span class="lab"><font color="Green" title="E36">A9</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:51_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_2"><span class="lab"><font color="Green" title="E36">A9</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">then </span><span class="kw">consider </span><font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c17">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:51_2"/><span class="lab"><font color="Green" title="E37">A10</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:51_2"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
<font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E9:51_2"/>
<font color="Maroon" title="c17">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/>
<a NAME="E10:51_2"/>
 ex <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">g1</font>,<font color="Olive" title="b2">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E4:51_2"><span class="lab"><font color="Green" title="E36">A9</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/>
<a NAME="E11:51_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_2"><span class="lab"><font color="Green" title="E37">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E12:51_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_2"><span class="lab"><font color="Green" title="E38">A11</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="E13:51_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:51"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">IG1</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">IG2</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:51_3"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_3_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c17">g2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_3_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">g1</font>,<font color="Maroon" title="c17">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
( <font color="Maroon" title="c16">g1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c17">g2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">g1</font> = <font color="Maroon" title="c16">g1</font>, <font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c17">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">g1</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">g2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a></span>;<br/>
<a NAME="E7:51_3_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
 not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:51_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1"><span class="lab"><font color="Green" title="E37">A13</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="E3:51_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51"><span class="lab"><font color="Green" title="E35">A8</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:51_3_1"/>
<font color="Maroon" title="c18">g1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">g2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E3:51_3_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E9:51_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/>
<a NAME="E10:51_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:51_3_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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">let </span><font color="Maroon" title="c15">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:51_3"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/>

<a NAME="E3:51_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A18</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:51_3"/><span class="lab"><font color="Green" title="E39">A19</font></span>: 
 not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c17">c2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:51_3"/><span class="lab"><font color="Green" title="E40">A20</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">c1</font>,<font color="Maroon" title="c17">c2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E7:51_3"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
( <font color="Maroon" title="c16">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c17">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3"><span class="lab"><font color="Green" title="E37">A17</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">c1</font> = <font color="Maroon" title="c16">c1</font>, <font color="Maroon" title="c19">c2</font> = <font color="Maroon" title="c17">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a></span>;<br/>
<a NAME="E9:51_3"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c18">c1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">c2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3"><span class="lab"><font color="Green" title="E38">A18</font></span></a>, <a class="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_3_2"/>
( ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> ) or ( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:51_3_2_1"/><span class="lab"><font color="Green" title="E43">A23</font></span>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">d2</font> = <font color="Maroon" title="c19">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">d1</font> = <font color="Maroon" title="c18">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_2_1"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E4:51_3_2_1"/>
<font color="Maroon" title="c21">d1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E9:51_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_2_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:51_3_2_2"/><span class="lab"><font color="Green" title="E43">A24</font></span>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_2_2"/><span class="lab"><font color="Green" title="E44">A25</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_2_2_1"/>
 not <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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/>

<a NAME="E2:51_3_2_2_1"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:51_3_2_2_1"/><span class="lab"><font color="Green" title="E45">A26</font></span>: 
<font color="Maroon" title="c20">b</font> <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"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:51_3_2_2_1"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_2_2_1"><span class="lab"><font color="Green" title="E45">A26</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E6:51_3_2_2_1"/>
<font color="Maroon" title="c20">b</font> <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"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</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="txt" href="#E4:51_3_2_2_1"><span class="lab"><font color="Green" title="E45">A26</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">then </span><span class="kw">consider </span><font color="Maroon" title="c21">b1</font>, <font color="Maroon" title="c22">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:51_3_2_2_1"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
<font color="Maroon" title="c21">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E9:51_3_2_2_1"/><span class="lab"><font color="Green" title="E48">A29</font></span>: 
<font color="Maroon" title="c22">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font>
 <span class="kw">and </span><br/><a NAME="E10:51_3_2_2_1"/><span class="lab"><font color="Green" title="E49">A30</font></span>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">b1</font>,<font color="Maroon" title="c22">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">b2</font> = <font color="Maroon" title="c22">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_2_2_1"><span class="lab"><font color="Green" title="E48">A29</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">b1</font> = <font color="Maroon" title="c21">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_2_2_1"><span class="lab"><font color="Green" title="E47">A28</font></span></a></span>;<br/>
<a NAME="E13:51_3_2_2_1"/>
(  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c24">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_2_2_1"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="ref" href="neckla_3.html#T29" target="_self" title="NECKLA_3:th.29">Th30</a>, <a class="ref" href="msualg_5.html#D1" title="MSUALG_5:def.1">MSUALG_5:def 1</a></span>;<br/>
<a NAME="E14:51_3_2_2_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c23">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_2_2_1"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E10:51_3_2_2_1"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="eqrel_1.html#T7" title="EQREL_1:th.7">EQREL_1:7</a></span>;<br/>
<a NAME="E15:51_3_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="neckla_3.html#T29" target="_self" title="NECKLA_3:th.29">Th30</a></span>;<br/>
<a NAME="E16:51_3_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_2_2_1"><span class="lab"><font color="Green" title="E48">A29</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="E17:51_3_2_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:51_3_2_2"/>
<font color="Maroon" title="c15">a</font> <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"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</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="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E1:51_3_2_2"><span class="lab"><font color="Green" title="E43">A24</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><a NAME="E4:51_3_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <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"><span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span>,<font color="Maroon" title="c10">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3"><span class="lab"><font color="Green" title="E38">A18</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="E5:51_3_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_2_2"><span class="lab"><font color="Green" title="E44">A25</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_3"><span class="kw">suppose </span></a><a NAME="E1:51_3_2_3"/><span class="lab"><font color="Green" title="E43">A31</font></span>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/><div class="add"><a NAME="E2:51_3_2_3"/><span class="lab"><font color="Green" title="E44">A32</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_2_3_1"/>
 not <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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/>

<a NAME="E2:51_3_2_3_1"/><span class="kw">then </span>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:51_3_2_3_1"/><span class="lab"><font color="Green" title="E45">A33</font></span>: 
<font color="Maroon" title="c20">b</font> <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="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E5:51_3_2_3_1"/>
<font color="Maroon" title="c20">b</font> <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="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_2_3_1"><span class="lab"><font color="Green" title="E45">A33</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">then </span><span class="kw">consider </span><font color="Maroon" title="c21">b1</font>, <font color="Maroon" title="c22">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:51_3_2_3_1"/><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<font color="Maroon" title="c21">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font>
 <span class="kw">and </span><br/><a NAME="E8:51_3_2_3_1"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
<font color="Maroon" title="c22">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E9:51_3_2_3_1"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">b1</font>,<font color="Maroon" title="c22">b2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">b1</font> = <font color="Maroon" title="c21">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_2_3_1"><span class="lab"><font color="Green" title="E46">A34</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">b2</font> = <font color="Maroon" title="c22">b2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_2_3_1"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>;<br/>
<a NAME="E12:51_3_2_3_1"/><span class="lab"><font color="Green" title="E49">A37</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c24">b2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_2_3_1"><span class="lab"><font color="Green" title="E47">A35</font></span></a>, <a class="ref" href="neckla_3.html#T29" target="_self" title="NECKLA_3:th.29">Th30</a></span>;<br/>
<a NAME="E13:51_3_2_3_1"/><span class="lab"><font color="Green" title="E50">A38</font></span>: 
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_5.html#D1" title="MSUALG_5:def.1">MSUALG_5:def 1</a></span>;<br/>
<a NAME="E14:51_3_2_3_1"/>
<font color="Maroon" title="c20">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_2_3_1"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E15:51_3_2_3_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c24">b2</font>,<font color="Maroon" title="c23">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_2_3_1"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E13:51_3_2_3_1"><span class="lab"><font color="Green" title="E50">A38</font></span></a>, <a class="ref" href="eqrel_1.html#T6" title="EQREL_1:th.6">EQREL_1:6</a></span>;<br/>
<a NAME="E16:51_3_2_3_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c23">b1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:51_3_2_3_1"><span class="lab"><font color="Green" title="E49">A37</font></span></a>, <a class="ref" href="eqrel_1.html#T7" title="EQREL_1:th.7">EQREL_1:7</a></span>;<br/>
<a NAME="E17:51_3_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="neckla_3.html#T29" target="_self" title="NECKLA_3:th.29">Th30</a></span>;<br/>
<a NAME="E18:51_3_2_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">A2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_2_3_1"><span class="lab"><font color="Green" title="E46">A34</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="E19:51_3_2_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:51_3_2_3"/>
<font color="Maroon" title="c15">a</font> <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="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="txt" href="#E1:51_3_2_3"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></span>;<br/><a NAME="E4:51_3_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">a</font> <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="c10">A2</font>,<span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3"><span class="lab"><font color="Green" title="E38">A18</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="E5:51_3_2_3"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_2_3"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0: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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2_4"><span class="kw">suppose </span></a><a NAME="E1:51_3_2_4"/><span class="lab"><font color="Green" title="E43">A39</font></span>: 
( <font color="Maroon" title="c18">c1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c19">c2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">d2</font> = <font color="Maroon" title="c19">c2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">d1</font> = <font color="Maroon" title="c18">c1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_2_4"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><a NAME="E4:51_3_2_4"/>
<font color="Maroon" title="c21">d1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c20">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E9:51_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E1:51_3_2_4"><span class="lab"><font color="Green" title="E43">A39</font></span></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></span>;<br/><span class="kw">hence </span><a NAME="E5:51_3_2_4"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E15:51"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c13">IG1</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">IG2</font> =  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:51_4"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_4_1"/><span class="lab"><font color="Green" title="E37">A40</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span></span><br/>

<span class="kw">assume </span><a NAME="E2:51_4_1"/>
 not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:51_4_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:51_4_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E1:51_4_1"><span class="lab"><font color="Green" title="E37">A40</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</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">let </span><font color="Maroon" title="c15">a</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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:51_4"/><span class="lab"><font color="Green" title="E37">A41</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:51_4_2"/>
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> or <font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_4"><span class="lab"><font color="Green" title="E37">A41</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2_1"><span class="kw">suppose </span></a><a NAME="E1:51_4_2_1"/><span class="lab"><font color="Green" title="E38">A42</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_4_2_1"/><span class="lab"><font color="Green" title="E39">A43</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:51_4_2_1"/><span class="lab"><font color="Green" title="E40">A44</font></span>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c19">w</font> = <font color="Maroon" title="c17">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p2">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4_2_1"><span class="lab"><font color="Green" title="E40">A44</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">v</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4_2_1"><span class="lab"><font color="Green" title="E40">A44</font></span></a></span>;<br/><a NAME="E7:51_4_2_1"/>
<font color="Maroon" title="c18">v</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4_2_1"><span class="lab"><font color="Green" title="E38">A42</font></span></a>, <a class="txt" href="#E3:51_4_2_1"><span class="lab"><font color="Green" title="E39">A43</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/><a NAME="E8:51_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/><span class="kw">hence </span><a NAME="E9:51_4_2_1"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_4_2_1"><span class="lab"><font color="Green" title="E39">A43</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_4_2_2"><span class="kw">suppose </span></a><a NAME="E1:51_4_2_2"/><span class="lab"><font color="Green" title="E38">A45</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</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="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c17">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:51_4_2_2"/><span class="lab"><font color="Green" title="E39">A46</font></span>: 
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">v</font>,<font color="Maroon" title="c17">w</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:51_4_2_2"/><span class="lab"><font color="Green" title="E40">A47</font></span>: 
( <font color="Maroon" title="c16">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> &amp; <font color="Maroon" title="c17">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">A2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="relset_1.html#T2" title="RELSET_1:th.2">RELSET_1:2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">v</font> = <font color="Maroon" title="c16">v</font>, <font color="Maroon" title="c19">w</font> = <font color="Maroon" title="c17">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4_2_2"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">u1</font> = <font color="Maroon" title="c18">v</font>, <font color="Maroon" title="c21">u2</font> = <font color="Maroon" title="c19">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_4_2_2"><span class="lab"><font color="Green" title="E40">A47</font></span></a></span>;<br/><a NAME="E7:51_4_2_2"/>
<font color="Maroon" title="c18">v</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c19">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_4_2_2"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E3:51_4_2_2"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/><a NAME="E8:51_4_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">u1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c21">u2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/><span class="kw">hence </span><a NAME="E9:51_4_2_2"/>
<font color="Maroon" title="c15">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_4_2_2"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E16:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A48</font></span>: 
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> (<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p3">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></span>;<br/>
<a NAME="E17:51"/>
 the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#T13" title="RELAT_2:th.13">RELAT_2:13</a></span>;<br/>
<a NAME="E18:51"/><span class="kw">then </span>
 not  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="eqrel_1.html#K3" title="EQREL_1:func.3">\/</a> <span class="p1">(<span class="default"> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> <a href="relset_1.html#K3" title="RELSET_1:func.3">~</a></span>)</span> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E30">A2</font></span></a></span>;<br/>
<a NAME="E19:51"/><span class="kw">then </span>
 not <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="rewrite1.html#R2" title="REWRITE1:pred.2">are_convertible_wrt</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D4" title="REWRITE1:def.4">REWRITE1:def 4</a></span>;<br/>
<a NAME="E20:51"/><span class="kw">then </span>
 not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_5.html#K1" title="MSUALG_5:func.1">EqCl</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_6.html#T41" title="MSUALG_6:th.41">MSUALG_6:41</a></span>;<br/>
<a NAME="E21:51"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="neckla_3.html#T29" target="_self" title="NECKLA_3:th.29">Th30</a></span>;<br/>
<a NAME="E22:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A49</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font> is   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E23:51"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> (<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <span class="p3">(<span class="default"><a href="neckla_3.html#K1" title="NECKLA_3:func.1">component</a> <font color="Maroon" title="c6">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c10">A2</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E8:51"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E10:51"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="ref" href="neckla_2.html#D2" title="NECKLA_2:def.2">NECKLA_2:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:51"/>
 ex <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a>   <a href="necklace.html#V1" title="NECKLACE:attr.1">symmetric</a>   <a href="necklace.html#V3" title="NECKLACE:attr.3">irreflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G2</font> &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="neckla_2.html#K1" title="NECKLA_2:func.1">union_of</a> (<font color="Olive" title="b1">G1</font>,<font color="Olive" title="b2">G2</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="txt" href="#E12:51"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E22:51"><span class="lab"><font color="Green" title="E38">A49</font></span></a>, <a class="txt" href="#E16:51"><span class="lab"><font color="Green" title="E37">A48</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/>


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