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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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">G</font>;<br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">F2</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">G</font>;<br/>







<i><font color="Green" title="E8">A1</font></i>: <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="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F2</font></span>)</span>
<b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</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">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></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"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
;<br/>
<div><i><font color="Green" title="E9">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_2"/>
<font color="Maroon" title="c5">k</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="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/><a NAME="E2:11_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:11_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">F1</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>;<br/><a NAME="E4:11_2"/><b>then </b><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c5">k</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">F1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:11_2"/><b>then </b><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c3">F1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Maroon" title="c5">k</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="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_3.html#T24" title="FINSEQ_3:th.24">FINSEQ_3:24</a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="group_5.html#E4:11_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="group_5.html#E5:11_2"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:11_3"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c5">k</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="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/><a NAME="E2:11_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>;<br/><a NAME="E3:11_3"/><b>then </b><i><font color="Green" title="E11">A6</font></i>: 
<font color="Maroon" title="c5">k</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">F2</font>
 <b>by </b><i><a class="txt" href="group_5.html#E1:11_3"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E4:11_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</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="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> &amp; <font color="Maroon" title="c5">k</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">F2</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a></i>;<br/><a NAME="E5:11_3"/><b>then </b>
<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="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</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="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">F1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">k</font></span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c5">k</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="group_5.html#E3:11_3"><i><font color="Green" title="E11">A6</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="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font>
<b>by </b><i><a class="txt" href="group_5.html#E3:11_3"><i><font color="Green" title="E11">A6</font></i></a>, <a class="ref" href="group_5.html#D1" target="_self" title="GROUP_5:def.1">Def1</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</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="c4">F2</font> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">F2</font></span>)</span> <a href="group_5.html#K1" title="GROUP_5:func.1">|^</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="group_5.html#E1:11"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="group_5.html#E2:11"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="finseq_3.html#T43" title="FINSEQ_3:th.43">FINSEQ_3:43</a></i>;<br/>


</div>
