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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:51"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> 0</span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> 0</span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:51"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:51_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">m</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Olive">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:51_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:51"/>
for <font color="Olive">n</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, <a class="txt" href="group_1.html#E1:51"><i><font color="Green">E40</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:51"/>
( <font color="Maroon">g</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1" target="_self">*</a> <font color="Maroon">g</font> implies <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font></span>)</span> )
 ;<br/>


</div>
