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<b>let </b><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green" title="E14">A1</font></i>: 
 <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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font>
 ;<br/>

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c3">a</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="E3:26"/>
not <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:26"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="aofa_000.html#K5" title="AOFA_000:func.5">orbit</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">y</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">n1</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="E7:26"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c5">y</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="c1">f</font>,<font color="Maroon" title="c6">n1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c3">a</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"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c6">n1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E5:26"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,$1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c4">A</font>;<br/>
<a NAME="E8:26"/><i><font color="Green" title="E18">A5</font></i>: 
 ex <font color="Olive" title="b1">n</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="txt" href="aofa_000.html#E5:26"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="aofa_000.html#E7:26"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">n</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="E10:26"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">n</font>]
 <b>and </b><br/><a NAME="E11:26"/><i><font color="Green" title="E20">A7</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="aofa_000.html#E8:26"><i><font color="Green" title="E18">A5</font></i></a>);<br/></i>
<div><i><font color="Green" title="E21">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><a NAME="E1:26_1"/>
<font color="Maroon" title="c8">i</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/><b>hence </b><a NAME="E2:26_1"/>
( <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> implies <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c8">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> )
 <b>by </b><i><a class="txt" href="aofa_000.html#E11:26"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c7">n</font>
;<br/>

<b>thus </b><a NAME="E13:26"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font>,<font color="Maroon" title="c2">g</font> <a href="aofa_000.html#K7" title="AOFA_000:func.7">iter</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</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="c1">f</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="aofa_000.html#E1:26"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="aofa_000.html#E2:26"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="aofa_000.html#E10:26"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="aofa_000.html#E12:26"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aofa_000.html#D7" target="_self" title="AOFA_000:def.7">Def7</a></i>;<br/>

<b>thus </b><a NAME="E14:26"/>
( <span class="p1">(<span class="default"><a href="funct_7.html#K9" title="FUNCT_7:func.9">iter</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a> <font color="Maroon" title="c4">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="c7">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="c1">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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">A</font> ) )
 <b>by </b><i><a class="txt" href="aofa_000.html#E10:26"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="aofa_000.html#E12:26"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>


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