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<b>let </b><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">l</font>, <font color="Maroon" title="c3">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>set </b><font color="Maroon" title="c4">g</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <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/>
<a NAME="E2:4"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="numeral1.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:4"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g'</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <b>by </b><i><a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<a NAME="E6:4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">g'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="E1:4_1"/>
<font color="Maroon" title="c6">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="c5">g'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c7">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font></span>)</span> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">o</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="c8">o</font> = <font color="Maroon" title="c7">o</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="numeral1.html#E3:4_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o</font>
<b>by </b><i><a class="txt" href="numeral1.html#E3:4_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">o</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T13" title="SEQ_1:th.13">SEQ_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c1">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c8">o</font></span>)</span>
<b>by </b><i><a class="ref" href="prepower.html#D1" title="PREPOWER:def.1">PREPOWER:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E6:4_1"/>
<font color="Maroon" title="c6">a</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="txt" href="numeral1.html#E3:4_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">l</font> <a href="prepower.html#K1" title="PREPOWER:func.1">GeoSeq</a> </span>)</span></span>)</span> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> <font color="Maroon" title="c3">m</font> is   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D8" title="ORDINAL1:def.8">ORDINAL1:def 8</a></i>;<br/>


</div>
