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

<b>reconsider </b><font color="Maroon">h</font> = 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  ;<br/>
<a NAME="E3:31"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> </span>)</span> &amp; ( for <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> holds <br/> <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  is    <a href="group_2.html#M1">Subgroup</a> of <font color="Olive">G1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1"/><i><font color="Green">E42</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> 
 
;<br/>
<a NAME="E2:31_1"/>
for <font color="Olive">G1</font> being   <a href="group_2.html#M1">Subgroup</a> of  <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> holds <br/> <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  is    <a href="group_2.html#M1">Subgroup</a> of <font color="Olive">G1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G1</font> be    <a href="group_2.html#M1">Subgroup</a> of  <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> ;<br/>

<b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 ;<br/>

<a NAME="E2:31_1_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_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><b>that </b><br/><a NAME="E1:31_1_1_1"/><i><font color="Green">E50</font></i>: 
<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_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:31_1_1_1"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 ;<br/>

<a NAME="E3:31_1_1_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> 
 
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E4:31_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> 
 
<b>by </b><i><a class="ref" href="group_2.html#D8">GROUP_2:def 8</a></i>;<br/>
<a NAME="E5:31_1_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="int_1.html#NM1">Integer</a> <b>by </b><i><a class="ref" href="int_1.html#D2">INT_1:def 2</a></i>;<br/>
<a NAME="E7:31_1_1_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">h''</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G1</font> <b>by </b><i/>;<br/>
<a NAME="E9:31_1_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">h''</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G1</font>
 
;<br/>
<a NAME="E10:31_1_1_1"/>
<font color="Maroon">h''</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T4">GROUP_4:4</a></i>;<br/>
<b>hence </b><a NAME="E11:31_1_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1_1"/>
 <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="group_2.html#T66">GROUP_2:66</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1"/>
( <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a> </span>)</span> &amp; ( for <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">h</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G1</font> holds <br/> <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  is    <a href="group_2.html#M1">Subgroup</a> of <font color="Olive">G1</font> ) )
 <b>by </b><i>, <a class="ref" href="group_2.html#D8">GROUP_2:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:31"/><b>then </b>
 <a href="group_2.html#K6">(Omega).</a> <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  <a href="group_2.html#R1">=</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">h</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="group_4.html#D5">GROUP_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:31"/>
 <a href="gr_cy_1.html#K3" target="_self">INT.Group</a>  <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K6" target="_self">@'</a> 1</span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="group_2.html#D8">GROUP_2:def 8</a></i>;<br/>


</div>
