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<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>assume </b><a NAME="E1:29"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">k</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">l</font> )
 ;<br/>

<a NAME="E2:29"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<a NAME="E3:29"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:29"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<font color="Maroon">m</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span> <a href="int_1.html#K6">mod</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="newton.html#T101">NEWTON:101</a>, </i>
<br/>.= 
0
<b>by </b><i/>
;<br/>
<a NAME="E7:29"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T69">NAT_1:69</a></i>;<br/>
<b>then </b><i><font color="Green">E43</font></i>: <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K6">mod</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">l</font>
<b>by </b><i><a class="ref" href="nat_1.html#T47">NAT_1:47</a></i>
;<br/>
<b>take </b>
<font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E9:29"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K5">div</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/>


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