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<div class="add">

<b>let </b><font color="Maroon">i1</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font>, <font color="Maroon">i2</font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:21_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</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">n</font> holds  <font color="Maroon">i1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Olive">l</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:21_1_2"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">l</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">n</font> holds  <font color="Maroon">i2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> <font color="Olive">l</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:21_1_2"/>
for <font color="Olive">k</font>, <font color="Olive">l</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">n</font> holds  <font color="Maroon">i1</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="binop_1.html#K2">.</a> <font color="Olive">k</font>,<font color="Olive">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font>, <font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="gr_cy_1.html#K1" target="_self">Segm</a> <font color="Maroon">n</font>;<br/>

<b>thus </b><font color="Maroon">i1</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">l</font> = 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">i2</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">k</font>,<font color="Maroon">l</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_1_2"/>
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


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