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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:25"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E2:25"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="gr_cy_1.html#K4" target="_self">addint</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:25"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green">E24</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green">E25</font></i></a>, <a class="ref" href="gr_cy_1.html#D5" target="_self">Def5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = 0 as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E6:25"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E2:25"><i><font color="Green">E25</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:25_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:25"><i><font color="Green">E24</font></i></a>, <a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a></i>;<br/>
<i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 0</span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:25"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:25"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E3:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>hence </b><a NAME="E6:25_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E4:25_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:25"/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K5" target="_self">INT.Group</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="group_1.html#T10">GROUP_1:10</a></i>;<br/>


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