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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="int_1.html#NM1">Integer</a>;<br/>









<b>assume </b><a NAME="E1:42"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E3:42"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E5:42"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="int_1.html#E1:42"><i><font color="Green">E20</font></i></a>, <a class="ref" href="int_1.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E6:42"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="int_1.html#E3:42"><i><font color="Green">E21</font></i></a>, <a class="txt" href="int_1.html#E5:42"><i><font color="Green">E22</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E2:42_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:42"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="int_1.html#R1" target="_self">are_congruent_mod</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self">Def3</a></i>;<br/>


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