<?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><br/> for <font color="Olive" title="b2">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="b2">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="Olive" title="b1">m</font> holds <br/><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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</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="Olive" title="b1">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</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"> 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/><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</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="Olive" title="b1">i</font></span><br/><b>let </b><font color="Maroon" title="c4">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="c4">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 <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c4">i</font> )</span><br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E43">AS</font></i>: 
<font color="Maroon" title="c4">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"> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c4">i</font></span><br/>

<a NAME="E2:61"/><i><font color="Green" title="E44">C</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> <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="E3:61"/><b>then </b><i><font color="Green" title="E45">CC</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> <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="E4:61"/><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/>
<b>then </b><i><font color="Green" title="E46">A1</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="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="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="E6:61"/>
<font color="Maroon" title="c3">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="int_6.html#E1:61"><i><font color="Green" title="E43">AS</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/>
<a NAME="E7:61"/><b>then </b><i><font color="Green" title="E47">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="c4">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="E8:61"/><i><font color="Green" title="E48">B1</font></i>: 
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c2">v</font>,<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="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_6.html#E5:61"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="int_6.html#E1:61"><i><font color="Green" title="E43">AS</font></i></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></i>;<br/>
<i><font color="Green" title="E49">D</font></i>:  <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> <font color="Maroon" title="c1">u</font>,<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="int_6.html#E3:61"><i><font color="Green" title="E45">CC</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 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> <font color="Maroon" title="c2">v</font>,<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="int_6.html#E2:61"><i><font color="Green" title="E44">C</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="E11:61"/><b>then </b>
( <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">u</font> <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="c4">i</font></span>)</span> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <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="c4">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="int_6.html#E9:61"><i><font color="Green" title="E49">D</font></i></a>, <a class="txt" href="int_6.html#E1:61"><i><font color="Green" title="E43">AS</font></i></a>, <a class="ref" href="int_6.html#D3" target="_self" title="INT_6:def.3">Df1</a></i>;<br/>
<a NAME="E12:61"/><b>then </b>
<span class="p1">(<span class="default"><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c2">v</font>,<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="c4">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="c4">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="c4">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_3.html#T14" title="INT_3:th.14">INT_3:14</a></i>;<br/>
<b>hence </b><a NAME="E13:61"/>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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="c4">i</font>
 <b>by </b><i><a class="txt" href="int_6.html#E8:61"><i><font color="Green" title="E48">B1</font></i></a>, <a class="txt" href="int_6.html#E7:61"><i><font color="Green" title="E47">D1</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/>


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