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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">ni</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">q</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c2">ni</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">ni</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> holds <br/>for <font color="Olive" title="b2">qc</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st <font color="Olive" title="b2">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> holds <br/>for <font color="Olive" title="b3">j</font>, <font color="Olive" title="b4">k</font>, <font color="Olive" title="b5">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b2">qc</font> &amp; <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b2">qc</font> &amp; <font color="Olive" title="b5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Olive" title="b2">qc</font> holds <br/><font color="Olive" title="b3">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b4">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Olive" title="b5">l</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">q</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">ni</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">ni</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> implies for <font color="Olive" title="b1">qc</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st <font color="Olive" title="b1">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> holds <br/>for <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">qc</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">qc</font> &amp; <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Olive" title="b1">qc</font> holds <br/><font color="Olive" title="b2">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b3">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Olive" title="b4">l</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E59">A1</font></span>: 
( <font color="Maroon" title="c2">ni</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">ni</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">qc</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>   st <font color="Olive" title="b1">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> holds <br/>for <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">qc</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Olive" title="b1">qc</font> &amp; <font color="Olive" title="b4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Olive" title="b1">qc</font> holds <br/><font color="Olive" title="b2">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b3">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Olive" title="b4">l</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">ttt</font> = <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K14" title="POLYNOM3:func.14">Polynom-Ring</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span>, <font color="Maroon" title="c6">p</font> being   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <span class="kw"> such that </span><br/><a NAME="E3:72"/><span class="lab"><font color="Green" title="E60">A2</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:72"/><span class="lab"><font color="Green" title="E61">A3</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; ( for <font color="Olive" title="b1">i</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</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#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/>( ( (  not <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> ) &amp; ( <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Olive" title="b1">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive" title="b1">i</font> ) ) ) )
 <span class="kw">and </span><br/><a NAME="E5:72"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
 <a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <span class="p2">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E59">A1</font></span></a>, <a class="ref" href="uniroots.html#T58" target="_self" title="UNIROOTS:th.58">Th58</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E63">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">i</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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="c5">f</font> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:72_1"/><span class="lab"><font color="Green" title="E64">A6</font></span>: 
<font color="Maroon" title="c7">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="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:72_1_1"/>
(  not <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> or ( <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:72_1_1_1"/>
(  not <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> or <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:72_1_1_1"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c7">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E1:72_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="72_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:72_1_1_2"/>
( <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">n</font> &amp; not <font color="Maroon" title="c7">i</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">ni</font> &amp; <font color="Maroon" title="c7">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:72_1_1_2"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> or <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c7">i</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72"><span class="lab"><font color="Green" title="E61">A3</font></span></a>, <a class="txt" href="#E1:72_1"><span class="lab"><font color="Green" title="E64">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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">let </span><font color="Maroon" title="c7">qc</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> implies for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Maroon" title="c7">qc</font> holds <br/><font color="Olive" title="b1">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b2">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Olive" title="b3">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:72"/>
<font color="Maroon" title="c7">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font>, <font color="Olive" title="b3">l</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Olive" title="b3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Maroon" title="c7">qc</font> holds <br/><font color="Olive" title="b1">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Olive" title="b2">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Olive" title="b3">l</font></span><br/>

<a NAME="E8:72"/><span class="kw">then </span>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">qc</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E60">A2</font></span></a>, <a class="txt" href="#E6:72"><span class="lab"><font color="Green" title="E63">A5</font></span></a>, <a class="ref" href="uniroots.html#T59" target="_self" title="UNIROOTS:th.59">Th59</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">m</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E10:72"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">qc</font>
 ;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c10">k</font>, <font color="Maroon" title="c11">l</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Maroon" title="c7">qc</font> implies <font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font> )</span><br/>

<span class="kw">assume </span><a NAME="E11:72"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> &amp; <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span>,<font color="Maroon" title="c7">qc</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">jc</font> = <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c13">lc</font> = <font color="Maroon" title="c11">l</font>, <font color="Maroon" title="c14">mc</font> = <font color="Maroon" title="c8">m</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D2" title="XCMPLX_0:def.2">XCMPLX_0:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">jcf</font> = <font color="Maroon" title="c12">jc</font>, <font color="Maroon" title="c16">lcf</font> = <font color="Maroon" title="c13">lc</font>, <font color="Maroon" title="c17">mcf</font> = <font color="Maroon" title="c14">mc</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></span>;<br/>
<a NAME="E14:72"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c7">qc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ,<font color="Maroon" title="c2">ni</font></span>)</span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <span class="p3">(<span class="default"><a href="uniroots.html#K5" title="UNIROOTS:func.5">cyclotomic_poly</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>,<font color="Maroon" title="c7">qc</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">qc</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:72"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E66">A9</font></span>: <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">lcf</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c15">jcf</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c17">mcf</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:72"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">l</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">m</font>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:72_3_1"/>
( <font color="Maroon" title="c11">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:72_3_1_1"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font></span><br/><div class="add"><a NAME="E2:72_3_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">l</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">j</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E66">A9</font></span></a></span>;<br/><a NAME="E3:72_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">l</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>;<br/><a NAME="E4:72_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c11">l</font> is  <a href="int_1.html#V1" title="INT_1:attr.1">integer</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_3_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="wsierp_1.html#T22" title="WSIERP_1:th.22">WSIERP_1:22</a></span>;<br/><a NAME="E5:72_3_1_1"/><span class="kw">then </span>
<span class="p1"><a href="int_1.html#K1" title="INT_1:func.1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c11">l</font></span>)</span></span><a href="int_1.html#K1" title="INT_1:func.1">/]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T47" title="INT_1:th.47">INT_1:47</a></span>;<br/><a NAME="E6:72_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</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="c9">j</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c11">l</font></span>)</span> <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:72"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="int_1.html#D7" title="INT_1:def.7">INT_1:def 7</a></span>;<br/><a NAME="E7:72_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72_3_1_1"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="ref" href="xcmplx_1.html#T90" title="XCMPLX_1:th.90">XCMPLX_1:90</a></span>;<br/><span class="kw">hence </span><a NAME="E8:72_3_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#D9" title="INT_1:def.9">INT_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="72_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:72_3_1_2"/>
<font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font></span><br/><div class="add"><a NAME="E2:72_3_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T75" title="INT_1:th.75">INT_1:75</a></span>;<br/><span class="kw">hence </span><a NAME="E3:72_3_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T16" title="INT_2:th.16">INT_2:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E17:72"/>
<font color="Maroon" title="c9">j</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> <font color="Maroon" title="c10">k</font> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <font color="Maroon" title="c11">l</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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