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

<b>let </b><font color="Maroon" title="c1">m</font> be    <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> ;<br/>

<a NAME="E1:64"/><i><font color="Green" title="E44">A1</font></i>: 
 <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></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="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="substlat.html#D4" title="SUBSTLAT:def.4">SUBSTLAT:def 4</a></i>;<br/>
<b>assume </b><a NAME="E2:64"/>
 <a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span><b> such that </b><br/><a NAME="E4:64"/><i><font color="Green" title="E45">A2</font></i>: 
(  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>  st  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Maroon" title="c2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font> ) )
 <b>by </b><i><a class="ref" href="lattice5.html#D2" title="LATTICE5:def.2">LATTICE5:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">EM</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="txt" href="heyting3.html#E1:64"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="substlat.html#T1" title="SUBSTLAT:th.1">SUBSTLAT:1</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="heyting3.html#T43" target="_self" title="HEYTING3:th.43">Th43</a></i>;<br/>
<a NAME="E7:64"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substlat.html#K1" title="SUBSTLAT:func.1">SubstitutionSet</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="heyting3.html#E1:64"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">a'</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">Mi</font> =  <a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c4">a'</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Cos</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> 1,<font color="Maroon" title="c1">m</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T31" target="_self" title="HEYTING3:th.31">Th31</a></i>;<br/>
<a NAME="E10:64"/><i><font color="Green" title="E47">A4</font></i>: 
 <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c6">Cos</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" title="c7">u</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:64_1"/>
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">n1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <b> such that </b><br/><a NAME="E3:64_1"/><i><font color="Green" title="E48">A5</font></i>: 
<font color="Maroon" title="c7">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self" title="HEYTING3:def.5">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" title="c9">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:64_1_1"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Cos</font>
 ;<br/><a NAME="E2:64_1_1"/><b>then </b><i><font color="Green" title="E49">A6</font></i>: 
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> 1,<font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:64_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E4:64_1_1"/><b>then </b>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> <font color="Maroon" title="c8">n1</font>,<font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_1_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/><b>hence </b><a NAME="E5:64_1_1"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">u</font> &amp; <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">d</font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_1"><i><font color="Green" title="E48">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:64_1"/>
<font color="Maroon" title="c6">Cos</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">u</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:64"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<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" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:64_2"/><b>then </b>
<font color="Maroon" title="c3">EM</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">Cos</font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="heyting3.html#E10:64"><i><font color="Green" title="E47">A4</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" title="HEYTING3:th.42">Th42</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">Mi</font> =  <a href="heyting2.html#K1" title="HEYTING2:func.1">Involved</a> <font color="Maroon" title="c4">a'</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="heyting3.html#E6:64"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="heyting3.html#T37" target="_self" title="HEYTING3:th.37">Th37</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">mX</font> = <span class="p1">(<span class="default"><a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c7">Mi</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">Sum</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">mX</font>,<font color="Maroon" title="c1">m</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T31" target="_self" title="HEYTING3:th.31">Th31</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>;<br/>
<a NAME="E15:64"/>
<font color="Maroon" title="c9">Sum</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</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" title="c11">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:64_3"/>
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">n</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> <b> such that </b><br/><a NAME="E3:64_3"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c11">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self" title="HEYTING3:def.5">Def5</a></i>;<br/>
<a NAME="E4:64_3"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Sum</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">t</font> &amp; <font color="Olive" title="b2">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">x</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" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:64_3_1"/><i><font color="Green" title="E49">A8</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Sum</font>
 ;<br/>

<a NAME="E2:64_3_1"/><b>then </b><i><font color="Green" title="E50">A9</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">mX</font>,<font color="Maroon" title="c1">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">mX</font> or <font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">mX</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" title="E51">A10</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">mX</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c14">y</font> =  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> <font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font>;<br/><b>thus </b><a NAME="E2:64_3_1_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_3"><i><font color="Green" title="E48">A7</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E3:64_3_1_1_1"/>
<font color="Maroon" title="c14">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_3_1"><i><font color="Green" title="E50">A9</font></i></a>, <a class="txt" href="heyting3.html#E1:64_3_1_1_1"><i><font color="Green" title="E51">A10</font></i></a>, <a class="txt" href="heyting3.html#E33"><i><font color="Green" title="E21">Lm7</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" title="E51">A11</font></i>: 
<font color="Maroon" title="c12">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">mX</font>
 ;<br/><div class="add"><b>take </b><font color="Maroon" title="c14">y</font> =  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">mX</font>,<font color="Maroon" title="c1">m</font>;<br/><b>thus </b><a NAME="E2:64_3_1_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:64_3"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="heyting3.html#E1:64_3_1_1_2"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E3:64_3_1_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">x</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:64_3_1"><i><font color="Green" title="E49">A8</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="c9">Sum</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">Th15</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:64"/><b>then </b><i><font color="Green" title="E48">A12</font></i>: 
 <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="heyting3.html#T34" target="_self" title="HEYTING3:th.34">Th34</a></i>;<br/>
<a NAME="E17:64"/><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<a NAME="E18:64"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</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" title="E50">A14</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 ;<br/>

<a NAME="E2:64_4"/><b>then </b><i><font color="Green" title="E51">A15</font></i>: 
<font color="Maroon" title="c9">Sum</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T24" title="YELLOW_0:th.24">YELLOW_0:24</a></i>;<br/>
<a NAME="E3:64_4"/>
 <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">mX</font>,<font color="Maroon" title="c1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Sum</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:64_4"/><i><font color="Green" title="E52">A16</font></i>: 
( <font color="Maroon" title="c11">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c11">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="heyting3.html#K2" title="HEYTING3:func.2">PFArt</a> <font color="Maroon" title="c8">mX</font>,<font color="Maroon" title="c1">m</font> )
 <b>by </b><i><a class="txt" href="heyting3.html#E2:64_4"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="heyting3.html#T15" target="_self" title="HEYTING3:th.15">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" title="c11">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or <font color="Maroon" title="c11">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="c11">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">n</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> <b> such that </b><br/><a NAME="E3:64_4_1_1"/><i><font color="Green" title="E53">A17</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c12">n</font> is <a href="abian.html#V1" title="ABIAN:attr.1">even</a> )
 <b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="heyting3.html#T33" target="_self" title="HEYTING3:th.33">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" title="b1">m'</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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>  st <br/>( <font color="Olive" title="b1">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font></span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green" title="E52">A16</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="heyting3.html#D2" target="_self" title="HEYTING3:def.2">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" title="b1">m'</font> being <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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>  st <br/>( <font color="Olive" title="b1">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">m'</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c13">m'</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <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> <b> such that </b><br/><a NAME="E3:64_4_1_1_1_1_1"/><i><font color="Green" title="E54">A18</font></i>: 
( <font color="Maroon" title="c13">m'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">m'</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</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" title="E53">A17</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1_1_1_1"><i><font color="Green" title="E54">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">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" title="E54">A19</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font></span>)</span>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c12">n</font>,<font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 ;<br/><div class="add"><a NAME="E2:64_4_1_1_1_1_2"/>
2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c8">mX</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">mX</font>
 <b>by </b><i><a class="ref" href="real_2.html#T144" title="REAL_2:th.144">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" title="E52">A16</font></i></a>, <a class="txt" href="heyting3.html#E3:64_4_1_1"><i><font color="Green" title="E53">A17</font></i></a>, <a class="txt" href="heyting3.html#E1:64_4_1_1_1_1_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="heyting3.html#T38" target="_self" title="HEYTING3:th.38">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" title="c11">g</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:64_4_1_2"/><b>then </b><i><font color="Green" title="E53">A20</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font> <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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E1:64_4"><i><font color="Green" title="E50">A14</font></i></a>, <a class="txt" href="heyting3.html#E5:64_4"><i><font color="Green" title="E52">A16</font></i></a>, <a class="ref" href="heyting3.html#T35" target="_self" title="HEYTING3:th.35">Th35</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">P1</font> =  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">m</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" title="HEYTING3:func.1">SubstPoset</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c1">m</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="heyting3.html#T28" target="_self" title="HEYTING3:th.28">Th28</a></i>;<br/><a NAME="E4:64_4_1_2"/><i><font color="Green" title="E54">A21</font></i>: 
 <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="heyting3.html#T30" target="_self" title="HEYTING3:th.30">Th30</a></i>;<br/><a NAME="E5:64_4_1_2"/>
 <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K5" title="HEYTING3:func.5">PFDrt</a> <font color="Maroon" title="c1">m</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D5" target="_self" title="HEYTING3:def.5">Def5</a></i>;<br/><a NAME="E6:64_4_1_2"/><b>then </b>
<font color="Maroon" title="c12">P1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 <b>by </b><i><a class="txt" href="heyting3.html#E16:64"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="lattice3.html#D8" title="LATTICE3:def.8">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" title="E53">A20</font></i></a>, <a class="txt" href="heyting3.html#E4:64_4_1_2"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="heyting3.html#T41" target="_self" title="HEYTING3:th.41">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" title="E50">A22</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E17:64"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/>
<a NAME="E20:64"/>
<font color="Maroon" title="c2">a</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c9">Sum</font>
 
<b>by </b><i><a class="txt" href="heyting3.html#E4:64"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="heyting3.html#E16:64"><i><font color="Green" title="E48">A12</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" title="E50">A22</font></i></a>, <a class="ref" href="orders_2.html#T30" title="ORDERS_2:th.30">ORDERS_2:30</a></i>;<br/>


</div>
