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<b>let </b><font color="Maroon">n1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><a NAME="E1:28_1_2"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> ) &amp;  ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> ) implies <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<b>consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:28_1_2_1"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">s</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>
<b>given </b><font color="Maroon">s</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E4:28_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E5:28_1_2_1"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font>
 ;<br/>

<b>given </b><font color="Maroon">p</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><a NAME="E6:28_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E7:28_1_2_1"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 ;<br/>

<a NAME="E8:28_1_2_1"/><i><font color="Green">E79</font></i>: 
 ex <font color="Olive">s1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">s1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:28_1_2_1"/>
 ex <font color="Olive">p1</font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <br/>( <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:28_1_2_1"/>
<font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font>
 <b>by </b><i>, , </i>;<br/>


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

<b>thus </b><a NAME="E2:28_1_2"/>
( not  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">n2</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">n1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n2</font> )
 ;<br/>


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