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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/><b>let </b><font color="Maroon" title="c4">b</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>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>consider </b><font color="Maroon" title="c6">k</font> being    <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> <b> such that </b><br/><a NAME="E3:4"/><i><font color="Green" title="E5">A2</font></i>: 
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font> )
 <b>by </b><i><a class="ref" href="int_1.html#T8" title="INT_1:th.8">INT_1:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> or <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font> )
 </span><b>by </b><i><a class="txt" href="group_8.html#E3:4"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="group_8.html#T3" target="_self" title="GROUP_8:th.3">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font>
 ;<br/><div class="add"><a NAME="E2:4_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="txt" href="group_8.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="group_2.html#T57" title="GROUP_2:th.57">GROUP_2:57</a></i>;<br/><a NAME="E3:4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="group_8.html#T3" target="_self" title="GROUP_8:th.3">Th3</a></i>;<br/><a NAME="E4:4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="ref" href="group_1.html#T72" title="GROUP_1:th.72">GROUP_1:72</a></i>;<br/><a NAME="E5:4_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T72" title="GROUP_1:th.72">GROUP_1:72</a></i>;<br/><a NAME="E6:4_1_2"/><b>then </b>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>;<br/><b>hence </b><a NAME="E7:4_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="group_8.html#E1:4_1_2"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="group_1.html#T71" title="GROUP_1:th.71">GROUP_1:71</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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