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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">h</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</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"> ( <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</span> implies  for <font color="Olive" title="b1">S</font> being    <a href="number14.html#M1" title="NUMBER14:mode.1">Sierp45FS</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Olive" title="b2">n</font>, <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Olive" title="b2">n</font>, <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 3 &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) holds <br/><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:54"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</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">  for <font color="Olive" title="b1">S</font> being    <a href="number14.html#M1" title="NUMBER14:mode.1">Sierp45FS</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font><br/>  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Olive" title="b2">n</font>, <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Olive" title="b2">n</font>, <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 3 &amp; (  for <font color="Olive" title="b3">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">S</font> holds <br/><font color="Olive" title="b2">n</font>,<font color="Olive" title="b1">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> ) holds <br/><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b2">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">r</font> =  <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c6">r0</font> =  <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c7">SGM</font> =  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">S</font> be    <a href="number14.html#M1" title="NUMBER14:mode.1">Sierp45FS</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>; <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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Olive" title="b1">n</font>, <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Olive" title="b1">n</font>, <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 3 &amp; (  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Olive" title="b1">n</font>,<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">i</font> ) holds <br/><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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"> ( <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a>  &amp; <font color="Maroon" title="c9">n</font>, <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Maroon" title="c9">n</font>, <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 3 &amp; (  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> ) implies <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font> <a href="zfmisc_1.html#R1" title="ZFMISC_1:pred.1">are_mutually_distinct</a> 
 <span class="kw">and </span><br/><a NAME="E3:54"/><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c9">n</font>, <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2
 <span class="kw">and </span><br/><a NAME="E4:54"/><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c9">n</font>, <a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 3
 <span class="kw">and </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E48">A5</font></span>: 
 for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> holds <br/><font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 ; <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"> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</a> </span><br/>

<span class="kw">assume </span><a NAME="E6:54"/><span class="lab"><font color="Green" title="E49">A6</font></span>: 
 not <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_coprime</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/>

<span class="kw">consider </span><font color="Maroon" title="c10">q</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a><span class="kw"> such that </span><br/><a NAME="E8:54"/><span class="lab"><font color="Green" title="E50">A7</font></span>: 
( <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> &amp; <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1"><a href="int_2.html#K1" title="INT_2:func.1">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span></span><a href="int_2.html#K1" title="INT_2:func.1">.|</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E49">A6</font></span></a>, <a class="ref" href="number14.html#T9" target="_self" onmouseover="rs('number14/T9')" onmouseout="rh()">Th9</a>, <a class="ref" href="pythtrip.html#D2" onmouseover="rs('pythtrip/D2')" onmouseout="rh()">PYTHTRIP:def 2</a></span>;<br/>
<a NAME="E9:54"/><span class="lab"><font color="Green" title="E51">A8</font></span>: 
( <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E50">A7</font></span></a>, <a class="ref" href="number14.html#T4" target="_self" onmouseover="rs('number14/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E10:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A9</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_5.html#T1" onmouseover="rs('int_5/T1')" onmouseout="rh()">INT_5:1</a></span>;<br/>
<a NAME="E11:54"/>
 <a href="number14.html#R4" title="NUMBER14:pred.4">two_or_more_are_odd_among</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:54_1"/>
 <a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>),<font color="Maroon" title="c9">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="int_1.html#T14" onmouseover="rs('int_1/T14')" onmouseout="rh()">INT_1:14</a></span>;<br/>
<a NAME="E2:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A10</font></span>: 
( <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 &amp; <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> 2 )
 
;<br/>
<a NAME="E3:54_1"/>
 <a href="number14.html#R4" title="NUMBER14:pred.4">two_or_more_are_odd_among</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K3" title="NUMBER14:func.3">numberR</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T33" target="_self" onmouseover="rs('number14/T33')" onmouseout="rh()">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_1"/>
 <a href="number14.html#R4" title="NUMBER14:pred.4">two_or_more_are_odd_among</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E53">A10</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/>


</div><span class="kw">end;</span></div>
<a NAME="E12:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E51">A8</font></span></a></span>;<br/>
<a NAME="E13:54"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:54_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="nat_d.html#T64" onmouseover="rs('nat_d/T64')" onmouseout="rh()">NAT_D:64</a></span>;<br/>
<a NAME="E2:54_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<a NAME="E3:54_2"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<a NAME="E4:54_2"/><span class="lab"><font color="Green" title="E57">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<a NAME="E5:54_2"/><span class="lab"><font color="Green" title="E58">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<a NAME="E6:54_2"/><span class="lab"><font color="Green" title="E59">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<a NAME="E7:54_2"/><span class="lab"><font color="Green" title="E60">A18</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n</font> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T66" onmouseover="rs('nat_d/T66')" onmouseout="rh()">NAT_D:66</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_2_1"><span class="kw">now </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"> ( (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> ) or ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> ) )</span></a><div class="add"><a NAME="E1:54_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3
 <span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T34" target="_self" onmouseover="rs('number14/T34')" onmouseout="rh()">Th34</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:54_2_1_1"/><span class="kw">then </span>
( (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> ) or ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> ) or (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_1"><span class="kw">case </span></a><a NAME="E1:54_2_1_1_1"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</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"> (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )</span><br/><div class="add"><a NAME="E2:54_2_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_2_1_1_1"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E5:54_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E6:54_2"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E7:54_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_2"><span class="kw">case </span></a><a NAME="E1:54_2_1_1_2"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</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"> (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )</span><br/><div class="add"><a NAME="E2:54_2_1_1_2"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_2_1_1_2"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp; 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E5:54_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E6:54_2"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E7:54_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_3"><span class="kw">case </span></a><a NAME="E1:54_2_1_1_3"/>
( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</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"> ( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )</span><br/><div class="add"><a NAME="E2:54_2_1_1_3"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_2_1_1_3"/>
( 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E5:54_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E6:54_2"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E7:54_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_1_1_4"><span class="kw">case </span></a><a NAME="E1:54_2_1_1_4"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</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"> (  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )</span><br/><div class="add"><a NAME="E2:54_2_1_1_4"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="number14.html#K4" title="NUMBER14:func.4">numberR0</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>)</span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">mod</a> 3 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</a></span>;<br/><span class="kw">hence </span><a NAME="E3:54_2_1_1_4"/>
(  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> &amp;  not 3 <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E2:54_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="txt" href="#E3:54_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="txt" href="#E4:54_2"><span class="lab"><font color="Green" title="E57">A15</font></span></a>, <a class="txt" href="#E5:54_2"><span class="lab"><font color="Green" title="E58">A16</font></span></a>, <a class="txt" href="#E6:54_2"><span class="lab"><font color="Green" title="E59">A17</font></span></a>, <a class="txt" href="#E7:54_2"><span class="lab"><font color="Green" title="E60">A18</font></span></a>, <a class="ref" href="int_1.html#T62" onmouseover="rs('int_1/T62')" onmouseout="rh()">INT_1:62</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E9:54_2"/>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">n</font> <a href="number14.html#R6" title="NUMBER14:pred.6">at_least_two_are_not_divisible_by</a> 3
 ; <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/>


</div><span class="kw">end;</span></div>
<a NAME="E14:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A19</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E51">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">h1</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c12">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <span class="kw"> such that </span><br/><a NAME="E16:54"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c11">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:54"/><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c12">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p1">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c11">h1</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:54"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">F</font>
 <span class="kw">and </span><br/><a NAME="E19:54"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> holds <br/>(  not <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp;  not <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> &amp;  not <font color="Maroon" title="c12">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="number14.html#D12" target="_self" onmouseover="rs('number14/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E20:54"/>
( <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E45">A2</font></span></a></span>;<br/>
<a NAME="E21:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A24</font></span>: 
 <a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="finseq_1.html#D14" onmouseover="rs('finseq_1/D14')" onmouseout="rh()">FINSEQ_1:def 14</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_3"><span class="kw">now </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">  not <font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:54_3"/>
<font color="Maroon" title="c10">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <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="E2:54_3"/><span class="kw">then </span>
 not  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  ...  &amp;  not <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><span class="kw">hence </span><a NAME="E3:54_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E14:54"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</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/></div><span class="kw">end;</span></div>
<a NAME="E23:54"/><span class="kw">then </span>
<font color="Maroon" title="c10">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E52">A9</font></span></a>, <a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="number14.html#T38" target="_self" onmouseover="rs('number14/T38')" onmouseout="rh()">Th38</a>, <a class="ref" href="int_2.html#T2" onmouseover="rs('int_2/T2')" onmouseout="rh()">INT_2:2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E25:54"/><span class="lab"><font color="Green" title="E60">A25</font></span>: 
<font color="Maroon" title="c13">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:54"/><span class="lab"><font color="Green" title="E61">A26</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:54"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c13">i</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:54"><span class="lab"><font color="Green" title="E60">A25</font></span></a></span>;<br/>
<a NAME="E28:54"/><span class="lab"><font color="Green" title="E62">A27</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14" title="FINSEQ_1:func.14">Sgm</a> <span class="p2">(<span class="default"><a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c4">h</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E16:54"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E17:54"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E18:54"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<a NAME="E29:54"/><span class="kw">then </span>
<font color="Maroon" title="c9">n</font>,<font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c10">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="txt" href="#E25:54"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="txt" href="#E26:54"><span class="lab"><font color="Green" title="E61">A26</font></span></a></span>;<br/>
<a NAME="E30:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A28</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c2">b</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c10">q</font>
 
;<br/>
<a NAME="E31:54"/>
(  not <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> &amp;  not <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> &amp;  not <font color="Maroon" title="c10">q</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="txt" href="#E16:54"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="txt" href="#E17:54"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E25:54"><span class="lab"><font color="Green" title="E60">A25</font></span></a>, <a class="txt" href="#E26:54"><span class="lab"><font color="Green" title="E61">A26</font></span></a>, <a class="txt" href="#E28:54"><span class="lab"><font color="Green" title="E62">A27</font></span></a>, <a class="txt" href="#E19:54"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:54"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E51">A8</font></span></a>, <a class="txt" href="#E30:54"><span class="lab"><font color="Green" title="E63">A28</font></span></a>, <a class="ref" href="number14.html#T19" target="_self" onmouseover="rs('number14/T19')" onmouseout="rh()">Th19</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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