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<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">b</font> <a href="group_1.html#NR1" target="_self">is_not_of_order_0</a> 
 ;<br/>

<a NAME="E2:11"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>;<br/>
<a NAME="E3:11"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> = <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E5:11"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">i</font>) ) )
 <b>from </b><i><a class="ref" href="finseq_1.html#S2">FINSEQ_1:sch 2</a>();<br/></i>
<a NAME="E6:11"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:11"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:11_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:11_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:11_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<b>set </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>;<br/>
<a NAME="E7:11_1"/>
<font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T78">NEWTON:78</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">k</font> = <font color="Maroon">m</font> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E9:11_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:11_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_1"><b>suppose </b></a><a NAME="E1:11_1_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:11_1_1_1"/>
(  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:11_1_1_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><font color="Maroon">a</font> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, , <a class="ref" href="group_1.html#T43">GROUP_1:43</a></i>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K6">|^</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, <a class="ref" href="group_1.html#D12">GROUP_1:def 12</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i>, </i>
;<br/><b>hence </b><a NAME="E5:11_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="11_1_1_2"><b>suppose </b></a><a NAME="E1:11_1_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:11_1_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="newton.html#T79">NEWTON:79</a></i>;<br/><a NAME="E3:11_1_1_2"/><b>then </b>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="group_1.html#K7">ord</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:11_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="group_1.html#K7">ord</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E5:11_1_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:11_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:11"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D14">GROUP_1:def 14</a><br/>


</div>
