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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>, <font color="Maroon">r</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">t</font> be    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t</font>;<br/><b>let </b><font color="Maroon">t1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of 3 <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><a NAME="E1:30"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:30"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:30"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:30"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E5:30"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:30"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:30"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:30"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E9:30"/><i><font color="Green">E46</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_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>assume </b><a NAME="E1:30_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">t1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">i</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E3:30_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:30_1"/><b>then </b>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">j</font> = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/><a NAME="E6:30_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="polynom1.html#K3">/.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">t1</font> <a href="polynom1.html#K3">/.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">t1</font> <a href="polynom1.html#K3">/.</a> <font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 3</span>)</span></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E7:30_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><font color="Maroon">t1</font> <a href="polynom1.html#K3">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">t1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon">t</font> <a href="polynom1.html#K3">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E9:30_1"/>
<span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="polynom3.html#T1" target="_self">Th1</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:30"/>
 <a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K7" target="_self">prodTuples</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">r</font>,<font color="Maroon">t</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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