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<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">i2</font>, <font color="Maroon" title="c3">i5</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>





<b>thus </b><a NAME="E1:45"/>
( <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font> implies <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i5</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">i0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">i5</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self" title="INT_1:def.3">Def3</a></i>;<br/>
<a NAME="E4:45_1"/>
( <font color="Maroon" title="c3">i5</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i5</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c4">i0</font> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1 is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> )
 
<b>by </b><i><a class="txt" href="int_1.html#E3:45_1"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:45_1"/>
<font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i5</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self" title="INT_1:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:45"/>
<font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i5</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">i0</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E4:45"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c3">i5</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">i5</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self" title="INT_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:45"/>
( <font color="Maroon" title="c3">i5</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">i2</font> &amp; <font color="Maroon" title="c4">i0</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 is   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> )
 
<b>by </b><i><a class="txt" href="int_1.html#E4:45"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:45"/>
<font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">i2</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i5</font>
 <b>by </b><i><a class="ref" href="int_1.html#D3" target="_self" title="INT_1:def.3">Def3</a></i>;<br/>


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