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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> 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="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>  implies <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">A</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span> ;<br/>
<span class="kw">assume </span><a NAME="E1:59"/><span class="lab"><font color="Green" title="E48">A1</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> 
 ; <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"> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">S</font> =  the    <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/>
<span class="kw">set </span><font color="Maroon" title="c6">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="c7">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">consider </span><font color="Maroon" title="c8">h</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>, <font color="Maroon" title="c9">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="E3:59"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c8">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>
 <span class="kw">and </span><br/><a NAME="E4:59"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c9">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="c8">h</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <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 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="c9">F</font>
 <span class="kw">and </span><br/><a NAME="E6:59"/>
 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>  the    <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> holds <br/>(  not <font color="Maroon" title="c9">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"> the    <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 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="c9">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"> the    <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 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="c9">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"> the    <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 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="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="number14.html#D12" target="_self" onmouseover="rs('number14/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">X</font> =  <a href="number14.html#K5" title="NUMBER14:func.5">PrimeDivisors&gt;3</a> <font color="Maroon" title="c8">h</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">m</font> = <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">u</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<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="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><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a>  the    <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/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">u</font> = <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<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="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><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a>  the    <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> as   <a href="valued_0.html#V5" title="VALUED_0:attr.5">integer-valued</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<a NAME="E8:59"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<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="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><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E9:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A5</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <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></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E10:59"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a></span>;<br/>
<a NAME="E11:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A6</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">F</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" onmouseover="rs('finseq_1/D7')" onmouseout="rh()">FINSEQ_1:def 7</a></span>;<br/>
<a NAME="E12:59"/><span class="lab"><font color="Green" title="E54">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E13:59"/><span class="lab"><font color="Green" title="E55">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E14:59"/><span class="lab"><font color="Green" title="E56">A9</font></span>: 
( <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="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<a NAME="E15:59"/>
<span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font> is  <a href="partfun3.html#V1" title="PARTFUN3:attr.1">positive-yielding</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rvsum_3.html#D1" onmouseover="rs('rvsum_3/D1')" onmouseout="rh()">RVSUM_3:def 1</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"> (  not <font color="Maroon" title="c14">n</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> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> or  not <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/>
<font color="Maroon" title="c14">n</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> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</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 <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<a NAME="E2:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A10</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E3:59_1"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="number14.html#T44" target="_self" onmouseover="rs('number14/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<a NAME="E4:59_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_1"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="ref" href="newton.html#D6" onmouseover="rs('newton/D6')" onmouseout="rh()">NEWTON:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:59_1"/>
 not <span class="p1">(<span class="default"><span class="p2"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">m</font> = <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default">2,3</span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">F</font> as   <a href="int_6.html#NM1" title="INT_6:NM.1">CR_Sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E56">A9</font></span></a>, <a class="ref" href="number14.html#T45" target="_self" onmouseover="rs('number14/T45')" onmouseout="rh()">Th45</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">Z</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">m</font> </span> } </span> ;<br/>
<a NAME="E17:59"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">m</font> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">m</font> </span> } </span>  or <font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_2"/>
<font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">z</font> where <font color="Olive" title="b1">z</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Olive" title="b1">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">m</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"> <font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">z</font> being   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:59_2"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
<font color="Maroon" title="c16">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">z</font>
 <span class="kw">and </span><br/><a NAME="E4:59_2"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="number14.html#R2" title="NUMBER14:pred.2">solves_CRT</a> <font color="Maroon" title="c13">u</font>,<font color="Maroon" title="c14">m</font>
 ;<br/>
<a NAME="E5:59_2"/>
<font color="Maroon" title="c13">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E6:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c17">z</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">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E54">A7</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="number14.html#T20" target="_self" onmouseover="rs('number14/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E7:59_2"/>
<font color="Maroon" title="c13">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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 class="kw">by</span> <span class="lab"><a class="ref" href="number14.html#T21" target="_self" onmouseover="rs('number14/T21')" onmouseout="rh()">Th21</a></span>;<br/>
<a NAME="E8:59_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon" title="c17">z</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">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E55">A8</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="number14.html#T20" target="_self" onmouseover="rs('number14/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E9:59_2"/>
 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>  the    <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> holds <br/><font color="Maroon" title="c17">z</font>, the    <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 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> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="59_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">i</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="c18">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>  the    <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> implies <font color="Maroon" title="c17">z</font>, the    <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 href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_2_1"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c18">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>  the    <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"> <font color="Maroon" title="c17">z</font>, the    <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 href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font></span><br/>

<a NAME="E2:59_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c13">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c14">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="number14.html#T22" target="_self" onmouseover="rs('number14/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<a NAME="E3:59_2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E51">A4</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="E4:59_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="number14.html#T23" target="_self" onmouseover="rs('number14/T23')" onmouseout="rh()">Th23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:59_2_1"/>
<font color="Maroon" title="c17">z</font>, the    <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 href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="int_1.html#R2" title="INT_1:pred.2">are_congruent_mod</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E2:59_2_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="number14.html#T23" target="_self" onmouseover="rs('number14/T23')" onmouseout="rh()">Th23</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="E10:59_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c2">b</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">z</font>,<font color="Maroon" title="c3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">z</font> <a href="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E6:59_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E8:59_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="number14.html#T43" target="_self" onmouseover="rs('number14/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:59_2"/>
<font color="Maroon" title="c16">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E57">A11</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>
<span class="kw">hence </span><a NAME="E18:59"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">n</font> where <font color="Olive" title="b1">n</font> is   <a href="xxreal_0.html#V2" title="XXREAL_0:attr.2">positive</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> : <font color="Maroon" title="c1">a</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b1">n</font>,<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="number03.html#R1" title="NUMBER03:pred.1">are_mutually_coprime</a>  </span> } </span>  is  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E52">A5</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E53">A6</font></span></a>, <a class="ref" href="number14.html#T31" target="_self" onmouseover="rs('number14/T31')" onmouseout="rh()">Th31</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>
