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<a NAME="E1:29_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:29_1_1"/><b>then </b><i><font color="Green" title="E17">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="aofa_000.html#D7" target="_self" title="AOFA_000:def.7">Def7</a></i>;<br/>
<a NAME="E3:29_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:29_1_1_1"/>
<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c6">b</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">b</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/>
<b>reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:29_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="aofa_000.html#D7" target="_self" title="AOFA_000:def.7">Def7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_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="E1:29_1_1_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> or  not <font color="Maroon" title="c2">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:29_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="aofa_000.html#D7" target="_self" title="AOFA_000:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E3:29_1_1_1_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_1_1_2"/>
not <font color="Maroon" title="c2">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c7">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:29_1_1_1_1_2"/><i><font color="Green" title="E19">A4</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> &amp; <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c3">A</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> holds <br/><span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="aofa_000.html#T10" target="_self" title="AOFA_000:th.10">Th10</a></i>;<br/><a NAME="E4:29_1_1_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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/><a NAME="E5:29_1_1_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funct_7.html#T76" title="FUNCT_7:th.76">FUNCT_7:76</a></i>;<br/><a NAME="E6:29_1_1_1_1_2"/><b>then </b>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c8">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="aofa_000.html#E3:29_1_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="aofa_000.html#E3:29_1_1_1_1_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="aofa_000.html#E4:29_1_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a>, <a class="ref" href="funct_7.html#T76" title="FUNCT_7:th.76">FUNCT_7:76</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_1_1"/>
<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c2">f</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:29_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="aofa_000.html#E2:29_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>


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