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

<b>reconsider </b><font color="Maroon" title="c1">h</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  <b>by </b><i><a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">h</font> = <font color="Maroon" title="c1">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  ;<br/>
<a NAME="E3:34"/>
( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">G1</font> being <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> holds <br/> <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_1"/>
for <font color="Olive" title="b1">G1</font> being   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> holds <br/> <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">G1</font> be    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font> implies  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">G1</font> )</span><br/>

<b>assume </b><a NAME="E1:34_1_1"/><i><font color="Green" title="E25">A1</font></i>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">G1</font></span><br/>

<a NAME="E2:34_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> </span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:34_1_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> </span>)</span>
 <b>and </b><br/><a NAME="E2:34_1_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c4">x</font> as   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:34_1_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="int_1.html#D2" title="INT_1:def.2">INT_1:def 2</a></i>;<br/>
<a NAME="E4:34_1_1_1"/>
<font color="Maroon" title="c2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">h''</font> = <font color="Maroon" title="c2">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G1</font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:34_1_1"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<a NAME="E6:34_1_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c6">h''</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">G1</font>
 
;<br/>
<a NAME="E7:34_1_1_1"/>
<font color="Maroon" title="c6">h''</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">h</font> <a href="group_1.html#K5" title="GROUP_1:func.5">|^</a> <font color="Maroon" title="c5">x'</font>
 
<b>by </b><i><a class="ref" href="group_4.html#T4" title="GROUP_4:th.4">GROUP_4:4</a></i>;<br/>
<b>hence </b><a NAME="E8:34_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E2:34_1_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="gr_cy_1.html#E6:34_1_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="gr_cy_1.html#T39" target="_self" title="GR_CY_1:th.39">Th39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_1_1"/>
 <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c3">G1</font>
 <b>by </b><i><a class="ref" href="group_2.html#T66" title="GROUP_2:th.66">GROUP_2:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_1"/>
( <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> </span>)</span> &amp; ( for <font color="Olive" title="b1">G1</font> being <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>   st <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">h</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> holds <br/> <a href="group_2.html#K6" title="GROUP_2:func.6">(Omega).</a> <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G1</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:34"/>
 <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K5" title="GR_CY_1:func.5">@'</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="group_4.html#D5" title="GROUP_4:def.5">GROUP_4:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
