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<div class="add">

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

<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon">G</font> is <a href="realset2.html#V3">trivial</a> implies (  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:18_1"/><i><font color="Green">E45</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D2" target="_self">GROUP_6:def 2</a><br/>

<a NAME="E2:18_1"/>
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>
<a NAME="E3:18_1"/><b>then </b>
 ex <font color="Olive">B</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">B</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">B</font> )
 
<b>by </b><i><a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>hence </b><a NAME="E4:18_1"/>
(  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a> )
 <b>by </b><i>, <a class="ref" href="card_1.html#T79">CARD_1:79</a>, <a class="ref" href="group_1.html#D14">GROUP_1:def 14</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:18"/><i><font color="Green">E46</font></i>: 
(  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a> )
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:18"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="group_1.html#D15">GROUP_1:def 15</a></i>;<br/>
<b>thus </b><a NAME="E5:18"/>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_6.html#D2" target="_self">GROUP_6:def 2</a><br/>


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