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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c4">j</font> implies <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> )</span><br/>

<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E26">AS1</font></i>: 
<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c4">j</font> or <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> )</span><br/>

<b>assume </b><a NAME="E2:33"/>
( <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c4">j</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">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span><br/>

<a NAME="E3:33"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c4">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font> )
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<a NAME="E4:33"/><b>then </b><i><font color="Green" title="E27">B</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c4">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="int_2.html#T27" title="INT_2:th.27">INT_2:27</a></i>;<br/>
<a NAME="E5:33"/>
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c3">i</font> <a href="int_2.html#NK3" title="INT_2:NK.3">lcm'</a> <font color="Maroon" title="c4">j</font>
 
<b>by </b><i><a class="txt" href="int_6.html#E1:33"><i><font color="Green" title="E26">AS1</font></i></a>, <a class="ref" href="int_6.html#T20" target="_self" title="INT_6:th.20">x1</a></i>;<br/>
<a NAME="E6:33"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c1">x</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="int_6.html#E4:33"><i><font color="Green" title="E27">B</font></i></a>, <a class="ref" href="int_2.html#T13" title="INT_2:th.13">INT_2:13</a></i>;<br/>
<b>hence </b><a NAME="E7:33"/>
<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</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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