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<a NAME="E1:105_1_1"/><i><font color="Green" title="E49">A1</font></i>: 
(  <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_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  #) &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="gr_cy_1.html#D4" title="GR_CY_1:def.4">GR_CY_1:def 4</a>, <a class="ref" href="numbers.html#T15" title="NUMBERS:th.15">NUMBERS:15</a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<a NAME="E2:105_1_1"/><b>then </b><i><font color="Green" title="E50">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:105_1_1_1"/><i><font color="Green" title="E51">A3</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ,<a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><a NAME="E2:105_1_1_1"/><b>then </b><i><font color="Green" title="E52">A4</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c2">i1</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c3">i2</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  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="txt" href="monoid_0.html#E1:105_1_1_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/><b>thus </b><a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> = 
<a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E1:105_1_1_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>
<br/>.= 
<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#D2" title="GR_CY_1:def.2">GR_CY_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:105_1_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="monoid_0.html#E1:105_1_1_1"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="monoid_0.html#E2:105_1_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:105_1_1"/><b>then </b><i><font color="Green" title="E51">A5</font></i>: 
 <a href="binop_2.html#K44" title="BINOP_2:func.44">addint</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a>  <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="txt" href="monoid_0.html#E1:105_1_1"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="monoid_0.html#E2:105_1_1"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<a NAME="E5:105_1_1"/>
 <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is    <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K3" title="MONOID_0:func.3">&lt;REAL,+&gt;</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:105_1_1_2"/>
<font color="Maroon">H<sub>2</sub></font>( <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a> ) <a href="funct_4.html#NR1" title="FUNCT_4:NR.1">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K3" title="MONOID_0:func.3">&lt;REAL,+&gt;</a> )
 <b>by </b><i><a class="txt" href="monoid_0.html#E4:105_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="gr_cy_1.html#D4" title="GR_CY_1:def.4">GR_CY_1:def 4</a>, <a class="ref" href="relat_1.html#T88" title="RELAT_1:th.88">RELAT_1:88</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">MONOID_0:def 23</a><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:105_1_1"/>
 <a href="gr_cy_1.html#K2" title="GR_CY_1:func.2">INT.Group</a>  is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="monoid_0.html#V13" title="MONOID_0:attr.13">invertible</a>  <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K3" title="MONOID_0:func.3">&lt;REAL,+&gt;</a> 
 ;<br/>


</div>
