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<b>let </b><font color="Maroon">i1</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i2</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i5</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i3</font> be   <a href="int_1.html#NM1">Integer</a>;<br/><b>let </b><font color="Maroon">i4</font> be   <a href="int_1.html#NM1">Integer</a>;<br/>









<b>assume </b><a NAME="E1:41"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">i1</font>,<font color="Maroon">i2</font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">i3</font>,<font color="Maroon">i4</font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">i5</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">i8</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">i5</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i2</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i9</font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:41"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">i5</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i9</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i3</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i4</font>
 <b>by </b><i><a class="txt" href="int_1.html#E1"><i><font color="Green">Lemma20</font></i></a>, </i>;<br/>
<a NAME="E6:41"/>
 ex <font color="Olive">i7</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">i5</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">i7</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i3</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i4</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i3</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i4</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i8</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i9</font></span>)</span>
 
<b>by </b><i><a class="txt" href="int_1.html#E1:41"><i><font color="Green">E21</font></i></a>, <a class="txt" href="int_1.html#E3:41"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:41_1"/>
 ex <font color="Olive">i7</font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">i5</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">i7</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i3</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i4</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:41"/>
<font color="Maroon">i1</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i3</font>,<font color="Maroon">i2</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i4</font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">i5</font>
 <b>by </b><i/>;<br/>


</div>
