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<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>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">T</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Olive">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> : verum </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:125_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Olive">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> : verum </span>}</span>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="125_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:125_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>
 ;<br/>

<b>then consider </b><font color="Maroon">s</font> being    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3" target="_self">*</a> <b> such that </b><br/><a NAME="E3:125_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E4:125_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 ;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font>, <font color="Maroon">t</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E6:125_1_1"/><i><font color="Green">E63</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> )
 <b>and </b><br/><a NAME="E7:125_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:125_1_1"/>
( <font color="Maroon">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> &amp; <font color="Maroon">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T36" target="_self">Th36</a>, <a class="txt" href="finseq_2.html#E4:125_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:125_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Olive">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_2.html#T34" target="_self">Th34</a>, <a class="ref" href="finseq_2.html#T37" target="_self">Th37</a></i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:125_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Olive">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Maroon">t</font> being    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:125_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font>
 ;<br/><a NAME="E5:125_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="finseq_2.html#K3" target="_self">*</a>  )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/><a NAME="E6:125_1"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> &amp; <font color="Maroon">z</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3" target="_self">*</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E7:125_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T34" target="_self">Th34</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:125"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">z</font> <a href="finseq_1.html#K8">^</a> <font color="Olive">t</font></span>)</span> where <font color="Olive">z</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">i</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font>, <font color="Olive">t</font> is    <a href="finseq_2.html#M2" target="_self">Element</a> of <font color="Maroon">j</font> <a href="finseq_2.html#K4" target="_self">-tuples_on</a> <font color="Maroon">D</font> : verum </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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