<?xml version="1.0"?>
<div class="add">

<b>reconsider </b><font color="Maroon" title="c3">n</font> = <font color="Maroon" title="c2">n</font> as  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a> <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <b>by </b><i><a class="txt" href="int_3.html#E1:43"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font>,   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> $2</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font>;<br/>
<a NAME="E2:43_1_1"/><i><font color="Green" title="E29">A2</font></i>: 
for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font>  ex <font color="Olive" title="b3">c</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">l</font>,<font color="Olive" title="b3">c</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">l</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">k'</font> = <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c7">l'</font> = <font color="Maroon" title="c5">l</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E2:43_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">k'</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">l'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">n</font>
 
<b>by </b><i><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">c</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">k'</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">l'</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c3">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font> <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">c</font>
;<br/>

<b>thus </b><a NAME="E4:43_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c5">l</font>,<font color="Maroon" title="c8">c</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:43_1_1"/>
 ex <font color="Olive" title="b1">c</font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font> st <br/>for <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c3">n</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">l</font>,<font color="Olive" title="b1">c</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">l</font>]
 
<b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="int_3.html#E2:43_1_1"><i><font color="Green" title="E29">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:43_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c2">n</font> st <br/>for <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> <font color="Maroon" title="c2">n</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">k</font>,<font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">l</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">n</font>
 ;<br/>


</div>
