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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k2</font> implies  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">t</font> )</span><br/>

<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E5">A1</font></i>: 
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c3">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">t</font></span><br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c1">n</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> 
 
;<br/>
<a NAME="E3:4"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:4"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="nat_d.html#T2" title="NAT_D:th.2">NAT_D:2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:4"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/>
<a NAME="E7:4"/>
<font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="nat_2.html#T26" title="NAT_2:th.26">NAT_2:26</a></i>;<br/>
<a NAME="E8:4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="idea_1.html#E6:4"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">t</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">t</font></span><br/>

<b>thus </b><a NAME="E10:4"/>
<font color="Maroon" title="c3">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">t</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:4"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="idea_1.html#E3:4"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="idea_1.html#E4:4"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="idea_1.html#E6:4"><i><font color="Green" title="E9">A5</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/>


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