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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">h</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">h</font> implies <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">h</font> )</span><br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> &amp; <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">h</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">h</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:15"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:15"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">u</font>
 <b>by </b><i><a class="txt" href="#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="nat_d.html#D3" target="_self" title="NAT_D:def.3">Def3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">h</font></span><br/><a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c1">i</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;  <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">i</font> )
 ;<br/><a NAME="E3:15_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">i</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">h</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:15_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="int_1.html#T86" title="INT_1:th.86">INT_1:86</a></i>;<br/><a NAME="E4:15_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">u</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">h</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">i</font></span>)</span></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">h</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:15"><i><font color="Green" title="E15">A3</font></i></a></i>;<br/><a NAME="E5:15_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c3">h</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font>
 <b>by </b><i><a class="txt" href="#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="nat_d.html#T1" target="_self" title="NAT_D:th.1">Th1</a></i>;<br/><a NAME="E6:15_1"/>
<font color="Maroon" title="c2">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E14">A2</font></i></a></i>;<br/><a NAME="E7:15_1"/><b>then </b>
<font color="Maroon" title="c3">h</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="#E4:15_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="#E5:15_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="nat_1.html#T18" title="NAT_1:th.18">NAT_1:18</a></i>;<br/><b>hence </b><a NAME="E8:15_1"/>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">h</font>
 <b>by </b><i><a class="txt" href="#E3:15_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="nat_d.html#T3" target="_self" title="NAT_D:th.3">Th3</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>hence </b><a NAME="E7:15"/>
<font color="Maroon" title="c1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">h</font>
 <b>by </b><i><a class="txt" href="#E3:15"><i><font color="Green" title="E14">A2</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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