<?xml version="1.0"?>
<div class="add">

<a NAME="E1:32_1_1"/>
 for <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c4">g</font></span>)</span> holds  <font color="Olive" title="b1">h</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c4">g</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">h</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> </span><br/>

<a NAME="E1:32_1_1_1"/>
<font color="Maroon" title="c5">h</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E3:32_1_1_1"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T45" title="GROUP_6:th.45">GROUP_6:45</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">n</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E5:32_1_1_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#D1" target="_self" title="GROUPP_1:def.1">Def1</a></span>;<br/>
<span class="lab"><font color="Green" title="E21">A3</font></span>: <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T37" title="GROUP_6:th.37">GROUP_6:37</a></span>
<br/>.= 
<font color="Maroon" title="c5">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1_1"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="group_8.html#T4" title="GROUP_8:th.4">GROUP_8:4</a></span>
;<br/>
<span class="lab"><font color="Green" title="E22">A4</font></span>: <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_1_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="group_1.html#T41" title="GROUP_1:th.41">GROUP_1:41</a></span>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#T31" title="GROUP_6:th.31">GROUP_6:31</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">h1</font> = <font color="Maroon" title="c5">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">H</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" title="GROUP_2:th.42">GROUP_2:42</a></span>;<br/>
<a NAME="E9:32_1_1_1"/>
<font color="Maroon" title="c8">h1</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c3">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E7:32_1_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="group_8.html#T4" title="GROUP_8:th.4">GROUP_8:4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E11:32_1_1_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
(  <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c8">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">r</font> &amp; <font color="Maroon" title="c9">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T2" target="_self" title="GROUPP_1:th.2">Th2</a>, <a class="ref" href="group_1.html#T44" title="GROUP_1:th.44">GROUP_1:44</a></span>;<br/>
<a NAME="E12:32_1_1_1"/>
 <a href="group_1.html#K6" title="GROUP_1:func.6">ord</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:32_1_1_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="group_8.html#T5" title="GROUP_8:th.5">GROUP_8:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:32_1_1_1"/>
<font color="Maroon" title="c5">h</font> is <font color="Maroon" title="c1">p</font> <a href="groupp_1.html#V1" title="GROUPP_1:attr.1">-power</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#D1" target="_self" title="GROUPP_1:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:32_1_1"/>
 <a href="group_6.html#K11" title="GROUP_6:func.11">Image</a> <font color="Maroon" title="c4">g</font> is <font color="Maroon" title="c1">p</font> <a href="group_10.html#V2" title="GROUP_10:attr.2">-group</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="groupp_1.html#T17" target="_self" title="GROUPP_1:th.17">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
