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<span class="kw">set </span><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font>;<br/>
<span class="kw">given </span><font color="Maroon" title="c11">q1</font>, <font color="Maroon" title="c12">q2</font>, <font color="Maroon" title="c13">q3</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><a NAME="E2:142_1_1"/><span class="lab"><font color="Green" title="E104">A1</font></span>: 
<font color="Maroon" title="c11">q1</font>,<font color="Maroon" title="c12">q2</font>,<font color="Maroon" title="c13">q3</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 <span class="kw">and </span><br/><a NAME="E3:142_1_1"/><span class="lab"><font color="Green" title="E105">A2</font></span>: 
( <font color="Maroon" title="c11">q1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> &amp; <font color="Maroon" title="c12">q2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> &amp; <font color="Maroon" title="c13">q3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font></span>)</span> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="number08.html#D8" target="_self" onmouseover="rs('number08/D8')" onmouseout="rh()">NUMBER08:def 8</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:142_1_1"/>
( ( <font color="Maroon" title="c11">q1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font> or <font color="Maroon" title="c11">q1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font> ) &amp; ( <font color="Maroon" title="c12">q2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font> or <font color="Maroon" title="c12">q2</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font> ) &amp; ( <font color="Maroon" title="c13">q3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c5">p1</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c7">m</font> or <font color="Maroon" title="c13">q3</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c6">p2</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c8">n</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:142_1_1"><span class="lab"><font color="Green" title="E105">A2</font></span></a>, <a class="ref" href="int_5.html#T7" onmouseover="rs('int_5/T7')" onmouseout="rh()">INT_5:7</a></span>;<br/>
<a NAME="E5:142_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> ) &amp; ( <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c12">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> ) &amp; ( <font color="Maroon" title="c13">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> or <font color="Maroon" title="c13">q3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T7" target="_self" onmouseover="rs('number08/T7')" onmouseout="rh()">Th7</a>, <a class="ref" href="nat_3.html#T5" onmouseover="rs('nat_3/T5')" onmouseout="rh()">NAT_3:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:142_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:142_1_1"><span class="lab"><font color="Green" title="E104">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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