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<b>let </b><font color="Maroon" title="c1">k</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/>

<b>thus </b><a NAME="E1:46"/>
 <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</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">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1"/><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">k</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1"/>
(  ex <font color="Olive" title="b1">m</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>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c2">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="Olive" title="b1">m</font>,<font color="Maroon" title="c1">k</font> ) or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E1:46_1"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/>
 ex <font color="Olive" title="b1">m</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>  st <br/>( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c2">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="Olive" title="b1">m</font>,<font color="Maroon" title="c1">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">m</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="E3:46_1_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
( <font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c2">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="c3">m</font>,<font color="Maroon" title="c1">k</font> )
 ;<br/><a NAME="E4:46_1_1_1"/>
<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <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="E5:46_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="heyting3.html#E3:46_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E6:46_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</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">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="heyting3.html#E3:46_1_1_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</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">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:46"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</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">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_2"/>
( <font color="Maroon" title="c2">x</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">k</font> or <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font> )
 </span><b>by </b><i><a class="txt" href="heyting3.html#E2:46"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_1"><b>suppose </b></a><a NAME="E1:46_2_1"/>
<font color="Maroon" title="c2">x</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">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_2_2"><b>suppose </b></a><a NAME="E1:46_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="heyting3.html#K3" title="HEYTING3:func.3">PFCrt</a> 1,<font color="Maroon" title="c1">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:46_2_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="heyting3.html#K4" title="HEYTING3:func.4">PFBrt</a> 1,<font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="heyting3.html#D4" target="_self" title="HEYTING3:def.4">Def4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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