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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>





<a NAME="E1:45"/><i><font color="Green" title="E24">A1</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>;<br/>
<a NAME="E2:45"/><b>then </b><i><font color="Green" title="E25">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:45"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>;<br/>
<a NAME="E4:45"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">q</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E28">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E2:45"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="polynom1.html#E4:45"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E6:45"/><i><font color="Green" title="E29">A6</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>;<br/>
<a NAME="E7:45"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom1.html#E5:45"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_2"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_2_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> or  ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> ) )
 </span><b>by </b><i><a class="txt" href="polynom1.html#E6:45"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="polynom1.html#E1:45_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_1"><b>suppose </b></a><a NAME="E1:45_2_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 ;<br/><div class="add"><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E6:45"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="polynom1.html#E1:45_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E1:45_2_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="polynom1.html#E1:45_2_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="polynom1.html#E1:45"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="polynom1.html#E1:45_2_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_2"><b>suppose </b></a><a NAME="E1:45_2_1_2"/>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:45_2_1_2"/><i><font color="Green" title="E32">A10</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">n</font> )
 ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font> = 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E6:45"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="polynom1.html#E1:45_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom1.html#E3:45_2_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">n</font>
<b>by </b><i><a class="txt" href="polynom1.html#E3:45_2_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="polynom1.html#D2" target="_self" title="POLYNOM1:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="polynom1.html#E2:45"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="polynom1.html#E3:45"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="polynom1.html#E3:45_2_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="finseq_4.html#T84" title="FINSEQ_4:th.84">FINSEQ_4:84</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:45"/>
<font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom1.html#E7:45"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="finseq_5.html#T13" title="FINSEQ_5:th.13">FINSEQ_5:13</a></i>;<br/>


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