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<b>set </b><font color="Maroon">u</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> : for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> </span>}</span> ;<br/>
<a NAME="E1:21_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> : for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:21_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> : for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x''</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:21_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x''</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x''</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:21_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">p</font>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E2:21_1_1_2"/>
for <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">p</font>
 ;<br/><b>hence </b><a NAME="E3:21_1_1_2"/>
 ex <font color="Olive">y</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="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">I</font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:21_1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<a NAME="E5:21_1_1"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">I'</font> = <font color="Maroon">I</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E7:21_1_1"/>
 <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I'</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:21_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp;  <a href="cqc_sim1.html#K10">min</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x'</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>, <font color="Maroon">y'</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:21_1_1_3"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/><a NAME="E2:21_1_1_3"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E3:21_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:21_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">y'</font>
 <b>by </b><i>, <a class="ref" href="cqc_sim1.html#D8">CQC_SIM1:def 8</a></i>;<br/><a NAME="E5:21_1_1_3"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">y'</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:21_1_1_3"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E12:21_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> : for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E13:21_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font> : for <font color="Olive">x'</font>, <font color="Olive">y'</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>  st <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> &amp; <font color="Olive">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">x'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">y'</font> </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div>
