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<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">R1</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">X</font>, <font color="Maroon">R2</font> be   <a href="relset_1.html#NM3">Relation</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R2</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:11_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font> <a href="lang1.html#K17">[*]</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:11_1"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E6:11_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>and </b><br/><a NAME="E7:11_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/><a NAME="E8:11_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/><a NAME="E9:11_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R2</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T31">RELAT_1:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E2:11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:11_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R2</font> <a href="lang1.html#K17">[*]</a> 
 <b>by </b><i>, , , , , <a class="ref" href="finseq_1.html#D16">FINSEQ_1:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11"/>
<font color="Maroon">R1</font> <a href="lang1.html#K17">[*]</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">R2</font> <a href="lang1.html#K17">[*]</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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