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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">h</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">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</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> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> $1</span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</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> 0 = 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</font>
<b>by </b><i><a class="ref" href="group_1.html#D8" target="_self" title="GROUP_1:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">G</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#T16" target="_self" title="GROUP_1:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> 0</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#D8" target="_self" title="GROUP_1:def.8">Def8</a></i>
;<br/>
<a NAME="E2:49"/><b>then </b><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<div><i><font color="Green" title="E38">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c2">h</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_1.html#E49"><i><font color="Green" title="E28">Lm2</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c4">n</font></span>)</span></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#T25" target="_self" title="GROUP_1:th.25">Th25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></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#T49" target="_self" title="GROUP_1:th.49">Th49</a></i>
;<br/><b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:49"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="group_1.html#E2:49"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="group_1.html#E3:49"><i><font color="Green" title="E38">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:49"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">h</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="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">h</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 ;<br/>


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