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<a NAME="E1:107_1_1"/><i><font color="Green">E28</font></i>: 
(  <a href="gr_cy_1.html#K3">INT.Group</a>  <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# <a href="numbers.html#K4">INT</a> ,<a href="binop_2.html#K44">addint</a>  #) &amp;  <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K44">addint</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K33">addreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a>, <a class="ref" href="numbers.html#T15">NUMBERS:15</a>, <a class="ref" href="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E2:107_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:107_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K4">INT</a> ,<a href="numbers.html#K4">INT</a> </span><a href="zfmisc_1.html#K2">:]</a></span>
 ;<br/><a NAME="E2:107_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a>  <a href="numbers.html#K4">INT</a>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">i2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  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="mcart_1.html#T10">MCART_1:10</a></i>;<br/><b>thus </b><a href="binop_2.html#K44">addint</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<a href="binop_2.html#K44">addint</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T4" target="_self">Th4</a></i>
<br/>.= 
<a href="binop_2.html#K33">addreal</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">i1</font>,<font color="Maroon">i2</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2">GR_CY_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K33">addreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K4">INT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="monoid_0.html#T4" target="_self">Th4</a>, <a class="ref" href="monoid_0.html#D20" target="_self">Def20</a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:107_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="binop_2.html#K44">addint</a>  <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#D16" target="_self">Def16</a>, <a class="ref" href="monoid_0.html#D20" target="_self">Def20</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E5:107_1_1"/>
 <a href="gr_cy_1.html#K3">INT.Group</a>  is    <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K3" target="_self">&lt;REAL,+&gt;</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:107_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="gr_cy_1.html#K3">INT.Group</a> ) <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K3" target="_self">&lt;REAL,+&gt;</a> )
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#D4">GR_CY_1:def 4</a>, <a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D23" target="_self">MONOID_0:def 23</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:107_1_1"/>
 <a href="gr_cy_1.html#K3">INT.Group</a>  is  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="monoid_0.html#V13" target="_self">invertible</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K3" target="_self">&lt;REAL,+&gt;</a> 
 ;<br/>


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