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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="scmfsa_2.html#M2" target="_self">FinSeq-Location</a>  =  <a href="scmfsa_2.html#K6" target="_self">fsloc</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>1</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="scmfsa_2.html#K1" target="_self">SCM+FSA</a> <b> such that </b><br/><a NAME="E2:15"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:15"/>
 <a href="numbers.html#K5">NAT</a> , <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a>  <a href="tarski.html#R2">are_equipotent</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">c<sub>1</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E1:15_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:15_1_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:15_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmfsa_2.html#E1:15_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="scmfsa_2.html#E2:15_1_1"><i><font color="Green">E7</font></i></a></i>;<br/>
 <a href="scmfsa_2.html#K6" target="_self">fsloc</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="scmfsa_2.html#E2:15"><i><font color="Green">E5</font></i></a></i>
<br/>.= 
 <a href="scmfsa_2.html#K6" target="_self">fsloc</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="scmfsa_2.html#E2:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scmfsa_2.html#E3:15_1_1"><i><font color="Green">E8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:15_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E2:15_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E3:15_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:15_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:15_1_2"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmfsa_2.html#E3:15_1_2"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E6:15_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6" target="_self">fsloc</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="scmfsa_2.html#E2:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scmfsa_2.html#E4:15_1_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a> 
 <b>by </b><i><a class="ref" href="scmfsa_2.html#D5" target="_self">Def5</a></i>;<br/>


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

<b>thus </b><a NAME="E4:15_1"/>
 <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:15_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a> 
 ;<br/>

<a NAME="E2:15_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is    <a href="scmfsa_2.html#M2" target="_self">FinSeq-Location</a> 
 
<b>by </b><i><a class="ref" href="scmfsa_2.html#D5" target="_self">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:15_1_3"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K6" target="_self">fsloc</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="scmfsa_2.html#T20" target="_self">Th20</a></i>;<br/>
<a NAME="E5:15_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E6:15_1_3"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="scmfsa_2.html#E2:15"><i><font color="Green">E5</font></i></a>, <a class="txt" href="scmfsa_2.html#E4:15_1_3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:15_1_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:15"/>
not  <a href="scmfsa_2.html#K3" target="_self">FinSeq-Locations</a>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="card_1.html#T68">CARD_1:68</a>, <a class="ref" href="card_4.html#T15">CARD_4:15</a></i>;<br/>


</div>
