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

<b>let </b><font color="Maroon" title="c1">u</font>, <font color="Maroon" title="c2">v</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="int_6.html#NM1" title="INT_6:NM.1">CR_Sequence</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Olive" title="b1">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">m</font></span>)</span></span>)</span>,<font color="Olive" title="b1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span><br/><b>let </b><font color="Maroon" title="c3">m</font> be   <a href="int_6.html#NM1" title="INT_6:NM.1">CR_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">m</font> implies  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> )</span><br/>



<b>assume </b><a NAME="E1:66"/><i><font color="Green" title="E48">A</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">m</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span><br/>

<a NAME="E2:66"/><i><font color="Green" title="E49">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="int_6.html#D3" target="_self" title="INT_6:def.3">Df1</a></i>;<br/>
<a NAME="E3:66"/><i><font color="Green" title="E50">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">m</font> holds <br/><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font>,<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<b>by </b><i><a class="ref" href="int_6.html#T32" target="_self" title="INT_6:th.32">lemat4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">z</font> =  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>;<br/>
<a NAME="E4:66"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="ref" href="int_6.html#D3" target="_self" title="INT_6:def.3">Df1</a></i>;<br/>
<a NAME="E5:66"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="ref" href="int_6.html#D3" target="_self" title="INT_6:def.3">Df1</a></i>;<br/>
<a NAME="E6:66"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="int_6.html#E5"><i><font color="Green" title="E5">length1</font></i></a></i>;<br/>
<a NAME="E7:66"/><b>then </b><i><font color="Green" title="E51">A3</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font> &amp;  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c3">m</font> )
 
<b>by </b><i><a class="ref" href="int_6.html#T31" target="_self" title="INT_6:th.31">lemat2</a></i>;<br/>
<a NAME="E8:66"/><i><font color="Green" title="E52">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">m</font> holds <br/> <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">m</font> implies  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<b>assume </b><a NAME="E1:66_1"/><i><font color="Green" title="E53">B0</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span><br/>

 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">m</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:66"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">m</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E3:66_1"/><b>then </b><i><font color="Green" title="E54">B1</font></i>: 
<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E53">B0</font></i></a>, <a class="ref" href="int_6.html#D3" target="_self" title="INT_6:def.3">Df1</a></i>;<br/>
<a NAME="E4:66_1"/>
 <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>,<font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E53">B0</font></i></a>, <a class="ref" href="int_6.html#T35" target="_self" title="INT_6:th.35">lemat3p</a></i>;<br/>
<a NAME="E5:66_1"/><b>then </b><i><font color="Green" title="E55">B3</font></i>: 
<span class="p1">(<span class="default"><a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p3">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>;<br/>
<a NAME="E6:66_1"/>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <font color="Maroon" title="c3">m</font>
 
<b>by </b><i><a class="txt" href="#E1:66_1"><i><font color="Green" title="E53">B0</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:66_1"/><b>then </b><i><font color="Green" title="E56">D1</font></i>: 
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="partfun3.html#D1" title="PARTFUN3:def.1">PARTFUN3:def 1</a></i>;<br/>
<a NAME="E8:66_1"/><b>then </b>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> is    <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> 
 
<b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<a NAME="E9:66_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_3.html#T13" title="INT_3:th.13">INT_3:13</a></i>;<br/>
<b>hence </b><a NAME="E10:66_1"/>
 <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font>,<span class="p1">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font></span>)</span>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="#E3:66_1"><i><font color="Green" title="E54">B1</font></i></a>, <a class="txt" href="#E7:66_1"><i><font color="Green" title="E56">D1</font></i></a>, <a class="txt" href="#E5:66_1"><i><font color="Green" title="E55">B3</font></i></a>, <a class="ref" href="int_3.html#T12" title="INT_3:th.12">INT_3:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E9:66"/>
 <a href="int_6.html#K2" title="INT_6:func.2">to_int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c1">u</font>,<font color="Maroon" title="c3">m</font></span>)</span> <a href="rvsum_1.html#K4" title="RVSUM_1:func.4">+</a> <span class="p2">(<span class="default"><a href="int_6.html#K1" title="INT_6:func.1">mod</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">m</font></span>)</span></span>)</span>,<font color="Maroon" title="c3">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="#E1:66"><i><font color="Green" title="E48">A</font></i></a>, <a class="txt" href="#E3:66"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="#E7:66"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="#E8:66"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="int_6.html#E35"><i><font color="Green" title="E32">Th3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
