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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> 0;<br/>
<a NAME="E1:33"/><i><font color="Green" title="E17">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">y</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:33_1"/>
<font color="Maroon" title="c1">x</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">y</font> = <font color="Maroon" title="c1">x</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> 0 as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a>  ;<br/>
<b>take </b>
<font color="Maroon" title="c2">y</font>
;<br/>

<b>thus </b><a NAME="E3:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> <b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E18">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="amistd_3.html#E1:33"><i><font color="Green" title="E17">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c1">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:33"/>
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E19">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> )
 ;<br/>

<b>assume </b><a NAME="E2:33_2"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c3">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="amistd_3.html#E1:33_2"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:33_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">y</font>] )
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:33"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:33_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E2:33_2"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="amistd_3.html#T16" target="_self" title="AMISTD_3:th.16">Th16</a></i>;<br/>


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

<b>thus </b><a NAME="E5:33"/><i><font color="Green" title="E19">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:33"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <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><br/>

<b>let </b><font color="Maroon" title="c2">a</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="E7:33"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_3.html#K1" title="AMISTD_3:func.1">TrivialInfiniteTree</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E9:33"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> 0
 ;<br/>
<a NAME="E10:33"/>
( <font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:33"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="amistd_3.html#E5:33"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="amistd_3.html#E9:33"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:33"/>
<font color="Maroon" title="c2">a</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> <font color="Maroon" title="c1">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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