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<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">R</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="algseq_1.html#R1" target="_self">is_at_least_length_of</a> <font color="Maroon">p</font>;<br/>
<a NAME="E1:16"/><i><font color="Green">E26</font></i>: 
 ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E2:16"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 
<b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:16"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">k</font> <a href="algseq_1.html#R1" target="_self">is_at_least_length_of</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="algseq_1.html#R1" target="_self">is_at_least_length_of</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 ;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E5:16"/>
( <font color="Maroon">k</font> <a href="algseq_1.html#R1" target="_self">is_at_least_length_of</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="algseq_1.html#R1" target="_self">is_at_least_length_of</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> ) )
 <b>by </b><i/>;<br/>


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