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<a NAME="E1:103_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:103_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T15" title="NUMBERS:th.15">NUMBERS:15</a>, <a class="ref" href="relat_1.html#T62" title="RELAT_1:th.62">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T96" title="ZFMISC_1:th.96">ZFMISC_1:96</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E52">A2</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:103_1_1_1"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">x</font></span><br/><a NAME="E2:103_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></span>;<br/><span class="kw">thus </span><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#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">i2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#D1" title="GR_CY_1:def.1">GR_CY_1:def 1</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E1:103_1_1_1"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="txt" href="#E2:103_1_1_1"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="ref" href="funct_1.html#T47" title="FUNCT_1:th.47">FUNCT_1:47</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:103_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:103_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_1"><span class="lab"><font color="Green" title="E51">A1</font></span></a>, <a class="txt" href="#E3:103_1_1"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<a NAME="E6:103_1_1"/><span class="kw">then </span>
<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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K2" title="MONOID_0:func.2">&lt;REAL,+&gt;</a> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gr_cy_1.html#D3" title="GR_CY_1:def.3">GR_CY_1:def 3</a>, <a class="ref" href="relat_1.html#T59" title="RELAT_1:th.59">RELAT_1:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:103_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="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="monoid_0.html#M2" title="MONOID_0:mode.2">SubStr</a> of  <a href="monoid_0.html#K2" title="MONOID_0:func.2">&lt;REAL,+&gt;</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="monoid_0.html#D23" target="_self" title="MONOID_0:def.23">Def23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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