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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>set </b><font color="Maroon">D</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<b>assume </b><a NAME="E1:25"/><i><font color="Green">E26</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T41">ZFMISC_1:41</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">t</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> = <font color="Maroon">y</font>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:25"/><i><font color="Green">E28</font></i>: 
(  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="algseq_1.html#S1" target="_self">ALGSEQ_1:sch 1</a>();<br/></i>
<a NAME="E7:25"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:25"/><b>then </b>
 <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:25"/><b>then </b>
 <a href="algseq_1.html#K3" target="_self">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E10:25"/>
 ex <font color="Olive">p</font> being  <a href="algseq_1.html#NM1">AlgSequence</a> of <font color="Maroon">R</font> st  <a href="algseq_1.html#K3" target="_self">len</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>


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