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

<span class="kw">set </span><font color="Maroon" title="c1">g</font> = <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">h</font> = 2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">S</font> =  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">T</font> =  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>;<br/>
<a NAME="E1:51"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T67" title="FUNCT_4:th.67">FUNCT_4:67</a></span>;<br/>
<a NAME="E2:51"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T67" title="FUNCT_4:th.67">FUNCT_4:67</a></span>;<br/>
<a NAME="E3:51"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<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="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> or <font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span></span><br/>

<a NAME="E2:51_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T18" title="FUNCT_4:th.18">FUNCT_4:18</a></span>;<br/>
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> = 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">1,3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></span>
;<br/>
<a NAME="E4:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E2:51_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a></span>;<br/>
<a NAME="E5:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_1"/>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E30">A6</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></span>
;<br/>
<a NAME="E5:51"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span> <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="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_1.html#D8" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3"><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"/>
( ( not  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1"><span class="kw">case </span></a><span class="kw">that </span><a NAME="E1:51_3_1"/>
not  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">and </span><br/><a NAME="E2:51_3_1"/>
not  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c6">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p3">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c7">A</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>;<br/><a NAME="E3:51_3_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 3</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"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 3</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</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_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:51_3_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></span>;<br/>
<a NAME="E6:51_3_1_1"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">3</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:51_3_1"/><span class="lab"><font color="Green" title="E33">A11</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</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_2"/><span class="lab"><font color="Green" title="E34">A12</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_2"/><span class="lab"><font color="Green" title="E35">A13</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:51_3_1_2"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:51_3_1_2"><span class="lab"><font color="Green" title="E34">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:51_3_1_2"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c8">B</font> = <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>;<br/><a NAME="E5:51_3_1"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</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_3"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_3"/><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:51_3_1_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></span>;<br/>
<a NAME="E6:51_3_1_3"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">2</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1_3"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51_3_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_1_3"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:51_3_1"/>
2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">3 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></span>;<br/><a NAME="E7:51_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A17</font></span>: 
2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2 is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>;<br/><a NAME="E8:51_3_1"/><span class="lab"><font color="Green" title="E36">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</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_4"/><span class="lab"><font color="Green" title="E37">A19</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_4"/><span class="lab"><font color="Green" title="E38">A20</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:51_3_1_4"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1_4"><span class="lab"><font color="Green" title="E37">A19</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_3_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_3_1_4"><span class="lab"><font color="Green" title="E38">A20</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A21</font></span>:  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="frechet.html#T39" title="FRECHET:th.39">FRECHET:39</a>, <a class="ref" href="partfun1.html#T138" title="PARTFUN1:th.138">PARTFUN1:138</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default">1,3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="enumset1.html#T45" title="ENUMSET1:th.45">ENUMSET1:45</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K2" title="ENUMSET1:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1,2,3</span><a href="enumset1.html#K2" title="ENUMSET1:func.2">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T112" title="ENUMSET1:th.112">ENUMSET1:112</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a>, <a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a></span>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>
;<br/><a NAME="E10:51_3_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default">1 <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 3</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D4" title="FUNCT_4:def.4">FUNCT_4:def 4</a></span>;<br/><a NAME="E11:51_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3 is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>;<br/><span class="kw">hence </span><a NAME="E12:51_3_1"/>
<font color="Maroon" title="c6">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="funct_4.html#T98" title="FUNCT_4:th.98">FUNCT_4:98</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  &amp;  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p3">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  ) )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">F</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_1"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="ref" href="funct_2.html#T31" title="FUNCT_2:th.31">FUNCT_2:31</a></span>;<br/><a NAME="E14:51_3_1"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c10">C</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span> or <font color="Maroon" title="c11">x</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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_3_1_6"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span></span><br/>

<a NAME="E2:51_3_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A24</font></span>: 
( <font color="Maroon" title="c11">x</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">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x</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">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x</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">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">x</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">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/>
<a NAME="E3:51_3_1_6"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
( 2 <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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> &amp; 3 <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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/>
<a NAME="E4:51_3_1_6"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> &amp; 1 <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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</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="#E2:51_3_1_6"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E3:51_3_1_6"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E6:51_3_1_6"/>
not <font color="Maroon" title="c12">x9</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="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_6_1"/>
<font color="Maroon" title="c12">x9</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="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:51_3_1_6_1"/>
<font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_6_1"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
( <font color="Maroon" title="c12">x9</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="c13">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c12">x9</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"><span class="p2">(<span class="default"><font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">i</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="necklace.html#T20" target="_self" title="NECKLACE:th.20">Th20</a></span>;<br/>
<a NAME="E5:51_3_1_6_1"/>
( ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c13">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c13">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_1_6"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="txt" href="#E4:51_3_1_6_1"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_3_1_6_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:51_3_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A27</font></span>: 
<font color="Maroon" title="c12">x9</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="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T50" title="SUBSET_1:th.50">SUBSET_1:50</a></span>;<br/>
<a NAME="E8:51_3_1_6"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51_3_1_6"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/>
<a NAME="E9:51_3_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_1_6"><span class="lab"><font color="Green" title="E42">A27</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">hence </span><a NAME="E10:51_3_1_6"/>
<font color="Maroon" title="c11">x</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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E15:51_3_1"/>
<font color="Maroon" title="c6">f</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p3">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_7"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:51_3_1_7"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">2,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E2:51_3_1_7"/><span class="kw">then </span>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E41">A29</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E4:51_3_1_7"/>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1_7"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E42">A30</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E6:51_3_1_7"/><span class="lab"><font color="Green" title="E43">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E7:51_3_1_7"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E44">A32</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E9:51_3_1_7"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:51_3_1_7"><span class="lab"><font color="Green" title="E43">A31</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E45">A33</font></span>: <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_3.html#D5" title="ORDERS_3:def.5">ORDERS_3:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E11:51_3_1_7"/>
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E12:51_3_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A34</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/>
<a NAME="E13:51_3_1_7"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c11">y</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T4" target="_self" title="NECKLACE:th.4">Th4</a></span>;<br/>
<a NAME="E15:51_3_1_7"/><span class="lab"><font color="Green" title="E47">A35</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:51_3_1_7"><span class="lab"><font color="Green" title="E46">A34</font></span></a>, <a class="ref" href="necklace.html#T25" target="_self" title="NECKLACE:th.25">Th25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E16:51_3_1_7"/><span class="lab"><font color="Green" title="E48">A36</font></span>: 
( <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> &amp; <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5"><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_1_7_5"/>
( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="txt" href="#E5:51"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_1"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_1"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_2"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E8:51_3_1_7"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E10:51_3_1_7"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_2"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_3"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_3"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_3"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_4"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_4"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_4"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_5"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_5"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_5"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E8:51_3_1_7"><span class="lab"><font color="Green" title="E44">A32</font></span></a>, <a class="txt" href="#E10:51_3_1_7"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_5"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_6"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_6"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_6"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_7"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_7"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E10:51_3_1_7"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E3:51_3_1_7"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_7"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_8"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_8"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_8"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_9"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_9"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_9"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_10"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_10"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_10"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E10:51_3_1_7"><span class="lab"><font color="Green" title="E45">A33</font></span></a>, <a class="txt" href="#E3:51_3_1_7"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_10"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_11"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_11"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_11"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_12"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_12"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_12"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E3:51_3_1_7"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E5:51_3_1_7"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_12"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_13"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_13"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_13"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_14"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_14"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_14"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_15"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_15"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_7_5_15"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K4" title="ENUMSET1:func.4">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">3,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="enumset1.html#K4" title="ENUMSET1:func.4">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_7"><span class="lab"><font color="Green" title="E48">A36</font></span></a>, <a class="txt" href="#E3:51_3_1_7"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E5:51_3_1_7"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="ref" href="enumset1.html#D4" title="ENUMSET1:def.4">ENUMSET1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_7_5_15"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_7_5_16"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_7_5_16"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_7_5_16"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_7"><span class="lab"><font color="Green" title="E47">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">take </span>
<font color="Maroon" title="c9">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Maroon" title="c9">F</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>  )</span><br/><span class="kw">thus </span><a NAME="E16:51_3_1"/>
<font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">F</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> </span><br/><span class="kw">thus </span><a NAME="E17:51_3_1"/>
<font color="Maroon" title="c9">F</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="orders_3.html#D5" title="ORDERS_3:def.5">ORDERS_3:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font> or for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_3_1_8"/>
<font color="Maroon" title="c10">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<a NAME="E2:51_3_1_8"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>;<br/>
<a NAME="E3:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>;<br/>
<a NAME="E4:51_3_1_8"/><span class="kw">then </span>
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 
<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="E5:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A38</font></span>: 
(  not <font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>;<br/>
<a NAME="E6:51_3_1_8"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> )
 
<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="E7:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A39</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">l</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">l</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:51_3_1_8"><span class="lab"><font color="Green" title="E40">A37</font></span></a>, <a class="ref" href="necklace.html#T19" target="_self" title="NECKLACE:th.19">Th19</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A40</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b1">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </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="lab"><font color="Green" title="E44">A41</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="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>
<br/>.= 
4
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T21" target="_self" title="NECKLACE:th.21">Th21</a></span>
;<br/>
<a NAME="E10:51_3_1_8"/><span class="kw">then </span>
( <font color="Maroon" title="c10">x</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  &amp; <font color="Maroon" title="c11">y</font> is  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T4" target="_self" title="NECKLACE:th.4">Th4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">i</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">j</font> = <font color="Maroon" title="c11">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E12:51_3_1_8"/><span class="lab"><font color="Green" title="E45">A42</font></span>: 
<font color="Maroon" title="c10">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 <span class="p1">(<span class="default"><a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p2">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span></span>)</span>
 
;<br/>
<a NAME="E13:51_3_1_8"/><span class="lab"><font color="Green" title="E46">A43</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4 </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:51_3_1_8"><span class="lab"><font color="Green" title="E42">A39</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E14:51_3_1_8"/><span class="lab"><font color="Green" title="E47">A44</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">i</font> &amp; <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_8_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font></span>
<div class="add"><span class="kw">assume </span><a NAME="E1:51_3_1_8_2_1"/><span class="lab"><font color="Green" title="E48">A45</font></span>: 
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_2_1_1"><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_1_8_2_1_1"/>
( <font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1_8_2_1"><span class="lab"><font color="Green" title="E48">A45</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_2_1_1_1"/><span class="lab"><font color="Green" title="E49">A46</font></span>: 
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_1_8_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_1_8"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_8_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:51_3_1_8"><span class="lab"><font color="Green" title="E46">A43</font></span></a>, <a class="txt" href="#E1:51_3_1_8_2_1_1_1"><span class="lab"><font color="Green" title="E49">A46</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_2_1_1_2"/><span class="lab"><font color="Green" title="E49">A47</font></span>: 
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:51_3_1_8_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 4
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_1_8"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E3:51_3_1_8_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:51_3_1_8"><span class="lab"><font color="Green" title="E43">A40</font></span></a>, <a class="txt" href="#E1:51_3_1_8_2_1_1_2"><span class="lab"><font color="Green" title="E49">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:51_3_1_8_2"/>
<font color="Maroon" title="c12">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_3_1_8"><span class="lab"><font color="Green" title="E41">A38</font></span></a>, <a class="txt" href="#E12:51_3_1_8"><span class="lab"><font color="Green" title="E45">A42</font></span></a>, <a class="ref" href="necklace.html#D9" target="_self" title="NECKLACE:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:51_3_1_8"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<span class="p1">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 2,3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T11" target="_self" title="NECKLACE:th.11">Th11</a></span>;<br/>
<a NAME="E16:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A49</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E17:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A50</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A51</font></span>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51_3_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:51_3_1_8"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_8"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E19:51_3_1_8"/><span class="lab"><font color="Green" title="E52">A52</font></span>: 
<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="necklace.html#T11" target="_self" title="NECKLACE:th.11">Th11</a></span>;<br/>
<a NAME="E20:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A53</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,3</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T65" title="FUNCT_4:th.65">FUNCT_4:65</a></span>;<br/>
<a NAME="E21:51_3_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A54</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E22:51_3_1_8"/><span class="lab"><font color="Green" title="E55">A55</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_3_1_8_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:51_3_1_8_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</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_8_4"/><span class="lab"><font color="Green" title="E56">A56</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:51_3_1_8_4"/><span class="lab"><font color="Green" title="E57">A57</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E5:51_3_1_8_4"/>
( <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:51_3_1_8"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="txt" href="#E3:51_3_1_8_4"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51_3_1_8_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_8"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E4:51_3_1_8_4"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E56">A58</font></span>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51_3_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:51_3_1_8"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="txt" href="#E21:51_3_1_8"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:51_3_1_8"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E24:51_3_1_8"/><span class="lab"><font color="Green" title="E57">A59</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:51_3_1_8"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E58">A60</font></span>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51_3_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:51_3_1_8"><span class="lab"><font color="Green" title="E57">A59</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:51_3_1_8"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<a NAME="E26:51_3_1_8"/><span class="lab"><font color="Green" title="E59">A61</font></span>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:51_3_1_8"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E60">A62</font></span>: <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2,3 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:51_3_1"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E7:51_3_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E8:51_3_1"><span class="lab"><font color="Green" title="E36">A18</font></span></a>, <a class="txt" href="#E4:51_3_1"><span class="lab"><font color="Green" title="E33">A11</font></span></a>, <a class="ref" href="necklace.html#T8" target="_self" title="NECKLACE:th.8">Th8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> 1,3</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:51_3_1_8"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="txt" href="#E26:51_3_1_8"><span class="lab"><font color="Green" title="E59">A61</font></span></a>, <a class="ref" href="funct_4.html#T17" title="FUNCT_4:th.17">FUNCT_4:17</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:51_3_1_8"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c14">a</font>, <font color="Maroon" title="c15">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> or  not <font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> or <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E28:51_3_1_8"/><span class="lab"><font color="Green" title="E61">A63</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">and </span><br/><a NAME="E29:51_3_1_8"/><span class="lab"><font color="Green" title="E62">A64</font></span>: 
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8"><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_1_8_8"/>
( ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:51_3_1_8"><span class="lab"><font color="Green" title="E44">A41</font></span></a>, <a class="ref" href="card_1.html#T90" title="CARD_1:th.90">CARD_1:90</a>, <a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_1"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_1"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_2"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_2"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_2"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_3"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_3"/><span class="lab"><font color="Green" title="E63">A65</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_3"/><span class="kw">then </span>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E25:51_3_1_8"><span class="lab"><font color="Green" title="E58">A60</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E18:51_3_1_8"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_3"><span class="lab"><font color="Green" title="E63">A65</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_3"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_4"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_4"/><span class="lab"><font color="Green" title="E63">A66</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_4"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E18:51_3_1_8"><span class="lab"><font color="Green" title="E51">A51</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E27:51_3_1_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_4"><span class="lab"><font color="Green" title="E63">A66</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_4"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_5"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_5"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_5"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_6"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_6"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_6"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_7"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_7"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_7"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_8"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_8"/><span class="lab"><font color="Green" title="E63">A67</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_8"/><span class="kw">then </span>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E27:51_3_1_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_8"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E23:51_3_1_8"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_8"><span class="lab"><font color="Green" title="E63">A67</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_8"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_9"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_9"/><span class="lab"><font color="Green" title="E63">A68</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_9"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E25:51_3_1_8"><span class="lab"><font color="Green" title="E58">A60</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_9"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E18:51_3_1_8"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_9"><span class="lab"><font color="Green" title="E63">A68</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_9"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_10"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_10"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_10"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_11"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_11"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_11"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_12"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_12"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_12"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_13"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_13"/><span class="lab"><font color="Green" title="E63">A69</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_13"/><span class="kw">then </span>
<font color="Maroon" title="c15">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E18:51_3_1_8"><span class="lab"><font color="Green" title="E51">A51</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_13"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E27:51_3_1_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_13"><span class="lab"><font color="Green" title="E63">A69</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_13"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_14"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_14"/><span class="lab"><font color="Green" title="E63">A70</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><a NAME="E2:51_3_1_8_8_14"/><span class="kw">then </span>
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:51_3_1_8"><span class="lab"><font color="Green" title="E61">A63</font></span></a>, <a class="txt" href="#E27:51_3_1_8"><span class="lab"><font color="Green" title="E60">A62</font></span></a></span>;<br/><a NAME="E3:51_3_1_8_8_14"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">a</font>,<font color="Maroon" title="c15">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:51_3_1_8"><span class="lab"><font color="Green" title="E62">A64</font></span></a>, <a class="txt" href="#E23:51_3_1_8"><span class="lab"><font color="Green" title="E56">A58</font></span></a>, <a class="txt" href="#E1:51_3_1_8_8_14"><span class="lab"><font color="Green" title="E63">A70</font></span></a>, <a class="ref" href="necklace.html#T26" target="_self" title="NECKLACE:th.26">Th26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:51_3_1_8_8_14"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_15"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_15"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_15"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_1_8_8_16"><span class="kw">suppose </span></a><a NAME="E1:51_3_1_8_8_16"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_1_8_8_16"/>
<font color="Maroon" title="c14">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:51_3_1_8"><span class="lab"><font color="Green" title="E47">A44</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_3_2"><span class="kw">case </span></a><a NAME="E1:51_3_2"/>
(  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:51_3_2"/>
(  <a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4 is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  <a href="necklace.html#K3" title="NECKLACE:func.3">ComplRelStr</a> <span class="p1">(<span class="default"><a href="necklace.html#K4" title="NECKLACE:func.4">Necklace</a> 4</span>)</span> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
