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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:64"/><i><font color="Green">E46</font></i>: 
 <a href="substlat.html#K1">SubstitutionSet</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E21"><i><font color="Green">Lemma11</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:64"/>
 <a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> is <a href="lattice3.html#V3">complete</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green">E47</font></i>: 
(  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> st  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="lattice5.html#D2">LATTICE5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="heyting3.html#E1:64"><i><font color="Green">E46</font></i></a>, <a class="ref" href="substlat.html#T1">SUBSTLAT:1</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="heyting3.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E7:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E1:64"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="heyting2.html#K1">Involved</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E10:64"/><i><font color="Green">E49</font></i>: 
 <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> 1,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:64_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E4:64_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self">Def4</a></i>;<br/><b>hence </b><a NAME="E5:64_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:64_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="heyting3.html#E10:64"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:64_2"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T42" target="_self">Th42</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="heyting2.html#K1">Involved</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="heyting3.html#E6:64"><i><font color="Green">E48</font></i></a>, <a class="ref" href="heyting3.html#T37" target="_self">Th37</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T31" target="_self">Th31</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<a NAME="E15:64"/>
<font color="Maroon">c<sub>9</sub></font> <a href="lattice3.html#R1">is_&lt;=_than</a>  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:64_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K4" target="_self">PFBrt</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E4:64_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:64_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:64_3_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_3_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_1"><b>suppose </b></a><a NAME="E1:64_3_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>14</sub></font> =  <a href="heyting3.html#K3" target="_self">PFCrt</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E2:64_3_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E3:64_3_1_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_3_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="heyting3.html#E1:64_3_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="heyting3.html#E35"><i><font color="Green">Lemma23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_3_1_1_2"><b>suppose </b></a><a NAME="E1:64_3_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>14</sub></font> =  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E2:64_3_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="heyting3.html#E1:64_3_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self">Def4</a></i>;<br/><b>thus </b><a NAME="E3:64_3_1_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:64_3_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:64_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:64"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green">E47</font></i></a>, <a class="ref" href="heyting3.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E17:64"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<a NAME="E18:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:64_4"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/>

<a NAME="E2:64_4"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24">YELLOW_0:24</a></i>;<br/>
<a NAME="E3:64_4"/>
 <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:64_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="tarski.html#R1">c=</a>  <a href="heyting3.html#K2" target="_self">PFArt</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_4"><i><font color="Green">E53</font></i></a>, <a class="ref" href="heyting3.html#T15" target="_self">Th15</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_4_1"/>
(  not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1"><b>suppose </b></a><a NAME="E1:64_4_1_1"/>
not <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_4_1_1"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> is <a href="abian.html#V1">even</a> )
 <b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="heyting3.html#T33" target="_self">Th33</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_4_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:64_4_1_1_1_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> ) or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self">Def2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:64_4_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:64_4_1_1_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> )
 ;<br/><b>thus </b><a NAME="E4:64_4_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_4_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:64_4_1_1_1_1_2"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:64_4_1_1_1_1_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="real_2.html#T144">REAL_2:144</a></i>;<br/><b>hence </b><a NAME="E3:64_4_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="heyting3.html#E1:64_4_1_1_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="heyting3.html#T38" target="_self">Th38</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:64_4_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="64_4_1_2"><b>suppose </b></a><a NAME="E1:64_4_1_2"/>
<font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:64_4_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:64_4"><i><font color="Green">E52</font></i></a>, <a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="heyting3.html#T35" target="_self">Th35</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> =  <a href="heyting3.html#K4" target="_self">PFBrt</a> 1,<font color="Maroon">c<sub>1</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T28" target="_self">Th28</a></i>;<br/><a NAME="E4:64_4_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="heyting3.html#K4" target="_self">PFBrt</a> 1,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="heyting3.html#T30" target="_self">Th30</a></i>;<br/><a NAME="E5:64_4_1_2"/>
 <a href="heyting3.html#K4" target="_self">PFBrt</a> 1,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting3.html#K5" target="_self">PFDrt</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:64_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="heyting3.html#E16:64"><i><font color="Green">E50</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/><b>hence </b><a NAME="E7:64_4_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_4_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="heyting3.html#E4:64_4_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="heyting3.html#T41" target="_self">Th41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:64"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E17:64"><i><font color="Green">E51</font></i></a>, <a class="ref" href="orders_2.html#D10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E20:64"/>
<font color="Maroon">c<sub>2</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green">E47</font></i></a>, <a class="txt" href="heyting3.html#E16:64"><i><font color="Green">E50</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:64"/>
contradiction
 <b>by </b><i><a class="txt" href="heyting3.html#E19:64"><i><font color="Green">E52</font></i></a>, <a class="ref" href="orders_2.html#T30">ORDERS_2:30</a></i>;<br/>


</div>
