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<span class="kw">set </span><font color="Maroon" title="c3">P</font> =  <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">PNR</font> =  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">CPNR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:37_2_1"/>
 <a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x9</font>, <font color="Maroon" title="c8">y9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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> <span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="group_6.html#D7" title="GROUP_6:def.7">GROUP_6:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x9</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y9</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x9</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y9</font>, <font color="Maroon" title="c11">xy</font> = <font color="Maroon" title="c7">x9</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">y9</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">Pxy</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">y9</font></span>)</span>, <font color="Maroon" title="c13">PxPy</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x9</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y9</font></span>)</span>, <font color="Maroon" title="c14">Px</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x9</font>, <font color="Maroon" title="c15">Py</font> = <span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y9</font> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E3:37_2_1_1"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c11">xy</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">x</font> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c10">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></span>;<br/>
<a NAME="E4:37_2_1_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
<font color="Maroon" title="c13">PxPy</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c14">Px</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c15">Py</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">b9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c16">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies <font color="Maroon" title="c12">Pxy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">PxPy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font> )</span><br/><span class="kw">assume </span><a NAME="E1:37_2_1_1_1"/>
<font color="Maroon" title="c16">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <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="c12">Pxy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">PxPy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c16">b9</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ;<br/><span class="kw">consider </span><font color="Maroon" title="c18">r</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="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:37_2_1_1_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E5:37_2_1_1_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
<font color="Maroon" title="c11">xy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">and </span><br/><a NAME="E6:37_2_1_1_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c18">r</font> holds <br/><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="polynom3.html#D11" title="POLYNOM3:def.11">POLYNOM3:def 11</a></span>;<br/><a NAME="E7:37_2_1_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">bn</font> = <font color="Maroon" title="c17">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">fcr</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 <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b4">i</font></span>)</span> &amp; <font color="Olive" title="b3">fcr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Olive" title="b3">fcr</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">fcr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; ( for <font color="Olive" title="b5">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">k</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="Olive" title="b3">fcr</font> holds <br/> ex <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b6">b1</font>,<font color="Olive" title="b7">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Olive" title="b3">fcr</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b6">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b7">b2</font></span>)</span> ) ) );<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">xybn</font> = <font color="Maroon" title="c11">xy</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">u</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 <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E11:37_2_1_1_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
<font color="Maroon" title="c13">PxPy</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">and </span><br/><a NAME="E12:37_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E13:37_2_1_1_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c21">u</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c21">u</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Px</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>;<br/><div><span class="lab"><font color="Green" title="E41">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">e9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c22">e9</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="c18">r</font> implies  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">e9</font>,<font color="Olive" title="b1">u</font>] ) )</span><br/><span class="kw">assume </span><a NAME="E1:37_2_1_1_1_1"/>
<font color="Maroon" title="c22">e9</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="c18">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">e9</font>,<font color="Olive" title="b1">u</font>] )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">e</font> = <font color="Maroon" title="c22">e9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">e</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c25">q</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c23">e</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c26">fcr</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 <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E5:37_2_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c24">p</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c25">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c26">fcr</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">fcr</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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="c26">fcr</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">b1</font>,<font color="Olive" title="b3">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c26">fcr</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">q</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></span>;<br/><a NAME="E6:37_2_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
<font color="Maroon" title="c26">fcr</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/><a NAME="E7:37_2_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">e</font>,<font color="Maroon" title="c26">fcr</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2_1_1_1_1"><span class="lab"><font color="Green" title="E42">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:37_2_1_1_1_1"/>
 ex <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c22">e9</font>,<font color="Olive" title="b1">u</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2_1_1_1_1"><span class="lab"><font color="Green" title="E43">A13</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><span class="kw">consider </span><font color="Maroon" title="c22">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font></span>)</span>,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E16:37_2_1_1_1"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</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="c18">r</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c22">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E14:37_2_1_1_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>);<br/></span><a NAME="E17:37_2_1_1_1"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c22">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ;<br/><a NAME="E18:37_2_1_1_1"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">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> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E19:37_2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c22">s</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="Olive" title="b1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><a NAME="E20:37_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">s</font> is  <a href="finseq_1.html#V1" title="FINSEQ_1:attr.1">FinSequence-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">s</font> = <font color="Maroon" title="c22">s</font> as    <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 <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:37_2_1_1_1"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</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="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><span class="kw"> such that </span><br/><a NAME="E23:37_2_1_1_1"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">bn</font>
 <span class="kw">and </span><br/><a NAME="E24:37_2_1_1_1"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
<font color="Maroon" title="c20">xybn</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">bn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2_1_1_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="hilbasis.html#T29" target="_self" title="HILBASIS:th.29">Th29</a></span>;<br/><a NAME="E25:37_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Pxy</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:37_2_1_1_1"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></span>;<br/><a NAME="E26:37_2_1_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E27:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A20</font></span>: 
1 <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="c23">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><span class="lab"><font color="Green" title="E49">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c25">p9</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c26">q9</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E2:37_2_1_1_1_2"/>
<font color="Maroon" title="c23">s</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> 1 is    <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 <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E3:37_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A22</font></span>: 
<font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 is    <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 <font color="Maroon" title="c1">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:37_2_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E4:37_2_1_1_1_2"/>
( <font color="Maroon" title="c25">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c26">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> )
 ;<br/><span class="kw">hence </span><a NAME="E5:37_2_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E27:37_2_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_2"><span class="lab"><font color="Green" title="E50">A22</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><span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">n1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">b1</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1  st <font color="Olive" title="b2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Olive" title="b3">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n1</font> &amp; <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> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <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> <font color="Olive" title="b3">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font> &amp; <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> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b3">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> );<br/><span class="kw">set </span><font color="Maroon" title="c25">t</font> =  <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font>;<br/><div><span class="lab"><font color="Green" title="E50">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">n19</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c26">n19</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="c21">u</font> implies  ex <font color="Olive" title="b1">n29</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n29</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">n19</font>,<font color="Olive" title="b1">n29</font>] ) )</span><br/><span class="kw">assume </span><a NAME="E1:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c26">n19</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="c21">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n29</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n29</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">n19</font>,<font color="Olive" title="b1">n29</font>] )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c26">n19</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_2_1_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>
;<br/><a NAME="E4:37_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E51">A24</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">b1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E6:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
<font color="Maroon" title="c28">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><a NAME="E7:37_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D11" title="PRE_POLY:def.11">PRE_POLY:def 11</a></span>;<br/><a NAME="E8:37_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E9:37_2_1_1_1_3"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">b1n</font> = <font color="Maroon" title="c28">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">p</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c31">q</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E12:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E55">A28</font></span>: 
( <font color="Maroon" title="c30">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c31">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E13:37_2_1_1_1_3"/>
<font color="Maroon" title="c29">b1n</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></span>;<br/><a NAME="E14:37_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c29">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c32">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E16:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
<font color="Maroon" title="c32">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A30</font></span>: 
<font color="Maroon" title="c29">b1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c32">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c33">i</font> = <font color="Maroon" title="c32">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A29</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c34">n2</font> = <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">i</font>;<br/><a NAME="E19:37_2_1_1_1_3"/><span class="lab"><font color="Green" title="E58">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E20:37_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E8:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E21:37_2_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><a NAME="E22:37_2_1_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E20:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E12:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A28</font></span></a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A33</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>
;<br/><a NAME="E24:37_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A34</font></span>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E20:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="pre_poly.html#T30" title="PRE_POLY:th.30">PRE_POLY:30</a></span>;<br/><a NAME="E25:37_2_1_1_1_3"/>
for <font color="Olive" title="b1">i9</font>, <font color="Olive" title="b2">n19</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">b19</font> being  <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1  st <font color="Olive" title="b2">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">n1</font> &amp; <font color="Olive" title="b3">b19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n19</font> &amp; <font color="Olive" title="b1">i9</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b19</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b3">b19</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font> &amp; <font color="Olive" title="b1">i9</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"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">b19</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c28">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c33">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Olive" title="b3">b19</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E16:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="txt" href="#E17:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="txt" href="#E8:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="txt" href="#E19:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="txt" href="#E23:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E60">A33</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><span class="kw">hence </span><a NAME="E26:37_2_1_1_1_3"/>
 ex <font color="Olive" title="b1">n29</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n29</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c26">n19</font>,<font color="Olive" title="b1">n29</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:37_2_1_1_1_3"><span class="lab"><font color="Green" title="E61">A34</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><span class="kw">consider </span><font color="Maroon" title="c26">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E31:37_2_1_1_1"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</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="c21">u</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E29:37_2_1_1_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>);<br/></span><a NAME="E32:37_2_1_1_1"/>
1 <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="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:37_2_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/><a NAME="E33:37_2_1_1_1"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T4" title="POLYNOM3:th.4">POLYNOM3:4</a></span>;<br/><a NAME="E34:37_2_1_1_1"/><span class="kw">then </span>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:37_2_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E28:37_2_1_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></span>;<br/><a NAME="E35:37_2_1_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T27" title="PRE_POLY:th.27">PRE_POLY:27</a></span>;<br/><a NAME="E36:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A36</font></span>: 
 <a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E37:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A37</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n19</font>, <font color="Maroon" title="c28">n29</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c27">n19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font> &amp; <font color="Maroon" title="c28">n29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font> &amp; <font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">n29</font> implies <font color="Maroon" title="c27">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n29</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E54">A38</font></span>: 
<font color="Maroon" title="c27">n19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font>
 <span class="kw">and </span><br/><a NAME="E2:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E55">A39</font></span>: 
<font color="Maroon" title="c28">n29</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font>
 <span class="kw">and </span><br/><a NAME="E3:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E56">A40</font></span>: 
<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c28">n29</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="c27">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n29</font></span><br/><a NAME="E4:37_2_1_1_1_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:37_2_1_1_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">n1</font> = <font color="Maroon" title="c27">n19</font>, <font color="Maroon" title="c30">n2</font> = <font color="Maroon" title="c28">n29</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E57">A41</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_2_1_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>
;<br/><a NAME="E7:37_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A42</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">b1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n1</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E9:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E59">A43</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E6:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E57">A41</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c32">b2</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">n2</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E11:37_2_1_1_1_4"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">b1n</font> = <font color="Maroon" title="c31">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c34">b2n</font> = <font color="Maroon" title="c32">b2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></span>;<br/><a NAME="E13:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E60">A44</font></span>: 
( <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></span>;<br/><a NAME="E14:37_2_1_1_1_4"/>
<font color="Maroon" title="c32">b2</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E59">A43</font></span></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><a NAME="E15:37_2_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c34">b2n</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></span>;<br/><a NAME="E16:37_2_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c34">b2n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">i2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E18:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E61">A45</font></span>: 
<font color="Maroon" title="c35">i2</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E62">A46</font></span>: 
<font color="Maroon" title="c34">b2n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c36">i2</font> = <font color="Maroon" title="c35">i2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E61">A45</font></span></a></span>;<br/><a NAME="E21:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E63">A47</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font> &amp; <font color="Maroon" title="c36">i2</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"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E61">A45</font></span></a>, <a class="txt" href="#E19:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E62">A46</font></span></a></span>;<br/><a NAME="E22:37_2_1_1_1_4"/>
<font color="Maroon" title="c31">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E58">A42</font></span></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><a NAME="E23:37_2_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c33">b1n</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></span>;<br/><a NAME="E24:37_2_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c33">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c37">i1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E64">A48</font></span>: 
<font color="Maroon" title="c37">i1</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E65">A49</font></span>: 
<font color="Maroon" title="c33">b1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c38">i1</font> = <font color="Maroon" title="c37">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E64">A48</font></span></a></span>;<br/><a NAME="E29:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E66">A50</font></span>: 
<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c29">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c38">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E1:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E26:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="txt" href="#E27:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E65">A49</font></span></a></span>;<br/><a NAME="E30:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E67">A51</font></span>: 
<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c30">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E61">A45</font></span></a>, <a class="txt" href="#E19:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E62">A46</font></span></a></span>;<br/><a NAME="E31:37_2_1_1_1_4"/><span class="lab"><font color="Green" title="E68">A52</font></span>: 
<font color="Maroon" title="c32">b2</font> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E32:37_2_1_1_1_4"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font> &amp; <font color="Maroon" title="c38">i1</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"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E1:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E26:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E64">A48</font></span></a>, <a class="txt" href="#E27:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E65">A49</font></span></a></span>;<br/><a NAME="E33:37_2_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A53</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c38">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c36">i2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="txt" href="#E29:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E66">A50</font></span></a>, <a class="txt" href="#E21:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E63">A47</font></span></a>, <a class="txt" href="#E30:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E67">A51</font></span></a>, <a class="txt" href="#E13:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E60">A44</font></span></a>, <a class="ref" href="polynom3.html#T22" title="POLYNOM3:th.22">POLYNOM3:22</a></span>;<br/><a NAME="E34:37_2_1_1_1_4"/>
<font color="Maroon" title="c31">b1</font> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c31">b1</font> = 
<font color="Maroon" title="c33">b1n</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c32">b2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E65">A49</font></span></a>, <a class="txt" href="#E19:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E62">A46</font></span></a>, <a class="txt" href="#E33:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E69">A53</font></span></a>, <a class="txt" href="#E31:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E68">A52</font></span></a>, <a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>
;<br/><span class="kw">hence </span><a NAME="E36:37_2_1_1_1_4"/>
<font color="Maroon" title="c27">n19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E6:37_2_1_1_1_4"><span class="lab"><font color="Green" title="E57">A41</font></span></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</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><a NAME="E39:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A54</font></span>: 
<font color="Maroon" title="c26">p</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></span>;<br/><a NAME="E40:37_2_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c27">n19</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</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="c27">n19</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> or <font color="Maroon" title="c27">n19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E55">A55</font></span>: 
<font color="Maroon" title="c27">n19</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 ; <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="c27">n19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">p</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">n1</font> = <font color="Maroon" title="c27">n19</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c29">i</font>, <font color="Maroon" title="c30">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E56">A56</font></span>: 
<font color="Maroon" title="c29">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="c23">s</font>
 <span class="kw">and </span><br/><a NAME="E5:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E57">A57</font></span>: 
<font color="Maroon" title="c30">j</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"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E58">A58</font></span>: 
<font color="Maroon" title="c28">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">j</font>
 <span class="kw">and </span><br/><a NAME="E7:37_2_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="ref" href="pre_poly.html#T29" title="PRE_POLY:th.29">PRE_POLY:29</a></span>;<br/>
<a NAME="E8:37_2_1_1_1_5"/>
<font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E4:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<a NAME="E9:37_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A59</font></span>: 
<font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> is    <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 <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">bj</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c30">j</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c32">bij</font> = <font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c33">p9</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c34">q9</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c29">i</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/>
<a NAME="E12:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E60">A60</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>;<br/>
<a NAME="E13:37_2_1_1_1_5"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E14:37_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A61</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E60">A60</font></span></a>, <a class="ref" href="xreal_1.html#T237" title="XREAL_1:th.237">XREAL_1:237</a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A62</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_2_1_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>
;<br/>
<a NAME="E16:37_2_1_1_1_5"/>
( <font color="Maroon" title="c33">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c34">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c29">i</font></span>)</span> )
 
;<br/>
<a NAME="E17:37_2_1_1_1_5"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E4:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="txt" href="#E9:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E59">A59</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E63">A63</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>
;<br/>
<a NAME="E19:37_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A64</font></span>: 
<font color="Maroon" title="c31">bj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/>
<a NAME="E20:37_2_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c31">bj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E21:37_2_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A65</font></span>: 
<font color="Maroon" title="c31">bj</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_5_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c35">k</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_5_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="E1:37_2_1_1_1_5_3_1"/>
( <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or  not <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_5_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1_1_5_3_1_1"/><span class="lab"><font color="Green" title="E66">A66</font></span>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_5_3_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_5_3_1_1_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="E1:37_2_1_1_1_5_3_1_1_1_1"/>
( <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> or  not <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_5_3_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1_1_5_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E67">A67</font></span>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font></span><br/><div class="add"><a NAME="E2:37_2_1_1_1_5_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A68</font></span>: 
<font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">bn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_5_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E67">A67</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
<font color="Maroon" title="c31">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:37_2_1_1_1_5_3_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E65">A65</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_5_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E68">A68</font></span></a>, <a class="ref" href="pre_poly.html#D11" title="PRE_POLY:def.11">PRE_POLY:def 11</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="37_2_1_1_1_5_3_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A69</font></span>: 
not <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font></span><br/><div class="add"><a NAME="E2:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="lab"><font color="Green" title="E68">A70</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E3:37_2_1_1_1_5_3_1_1_1_1_2"/>
<font color="Maroon" title="c29">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2_1_1_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E4:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A71</font></span>: 
<font color="Maroon" title="c29">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c2">n</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></span>
;<br/><a NAME="E6:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">n</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_5_3_1_1"><span class="lab"><font color="Green" title="E66">A66</font></span></a>, <a class="txt" href="#E1:37_2_1_1_1_5_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E7:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A72</font></span>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E8:37_2_1_1_1_5_3_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>;<br/><span class="kw">hence </span><a NAME="E9:37_2_1_1_1_5_3_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:37_2_1_1_1_5_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A72</font></span></a>, <a class="txt" href="#E2:37_2_1_1_1_5_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A70</font></span></a>, <a class="txt" href="#E4:37_2_1_1_1_5_3_1_1_1_1_2"><span class="lab"><font color="Green" title="E69">A71</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</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="E3:37_2_1_1_1_5_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_5_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1_1_5_3_1_2"/><span class="lab"><font color="Green" title="E66">A73</font></span>: 
not <font color="Maroon" title="c35">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:37_2_1_1_1_5_3_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:37_2_1_1_1_5_3_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_2_1_1_1_5_3_1_2"><span class="lab"><font color="Green" title="E66">A73</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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>
<a NAME="E23:37_2_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D11" title="PRE_POLY:def.11">PRE_POLY:def 11</a></span>;<br/>
<a NAME="E24:37_2_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E26:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E66">A74</font></span>: 
<font color="Maroon" title="c35">k</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E67">A75</font></span>: 
<font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E28:37_2_1_1_1_5"/><span class="lab"><font color="Green" title="E68">A76</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:37_2_1_1_1"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E29:37_2_1_1_1_5"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E64">A64</font></span></a>, <a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></span>;<br/>
<a NAME="E30:37_2_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c31">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p2">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c30">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E5:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="txt" href="#E26:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E66">A74</font></span></a>, <a class="txt" href="#E27:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E67">A75</font></span></a>, <a class="txt" href="#E15:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E62">A62</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:37_2_1_1_1_5"/>
<font color="Maroon" title="c27">n19</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="txt" href="#E26:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E66">A74</font></span></a>, <a class="txt" href="#E28:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E68">A76</font></span></a>, <a class="txt" href="#E15:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E62">A62</font></span></a>, <a class="txt" href="#E14:37_2_1_1_1_5"><span class="lab"><font color="Green" title="E61">A61</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</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><a NAME="E41:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A77</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c26">p</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/><a NAME="E42:37_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <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="c23">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></span>;<br/><a NAME="E43:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A78</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><div><span class="lab"><font color="Green" title="E57">A79</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c27">k</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="c18">r</font> implies <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c28">p</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>, <font color="Maroon" title="c29">q</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c27">k</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">pq9</font> = <font color="Maroon" title="c28">p</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c29">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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="polynom1.html#K13" title="POLYNOM1:func.13">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><span class="kw">assume </span><a NAME="E3:37_2_1_1_1_6"/><span class="lab"><font color="Green" title="E58">A80</font></span>: 
<font color="Maroon" title="c27">k</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="c18">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">sk</font> = <font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c32">sk</font> = <font color="Maroon" title="c31">sk</font> as    <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 <font color="Maroon" title="c1">R</font> ;<br/><span class="lab"><font color="Green" title="E59">A81</font></span>: <font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c27">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c27">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:37_2_1_1_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E58">A80</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c30">pq9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>
;<br/><a NAME="E7:37_2_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> &amp; <font color="Maroon" title="c23">s</font> <a href="pre_poly.html#K9" title="PRE_POLY:func.9">/.</a> <font color="Maroon" title="c27">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E43:37_2_1_1_1"><span class="lab"><font color="Green" title="E56">A78</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E58">A80</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c32">sk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:37_2_1_1_1"><span class="lab"><font color="Green" title="E56">A78</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E58">A80</font></span></a>, <a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">p</font> <a href="polynom1.html#K12" title="POLYNOM1:func.12">*'</a> <font color="Maroon" title="c29">q</font></span>)</span> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c19">bn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E58">A80</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:37_2_1_1_1"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E6:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E59">A81</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1_1_6"><span class="lab"><font color="Green" title="E58">A80</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>
;<br/> <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="lab"><font color="Green" title="E58">A82</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c27">n1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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"> ( <font color="Maroon" title="c27">n1</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="c21">u</font> implies <font color="Maroon" title="c21">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n1</font></span>)</span> )</span><br/><a NAME="E1:37_2_1_1_1_7"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c28">b19</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ;<br/><span class="kw">assume </span><a NAME="E3:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E59">A83</font></span>: 
<font color="Maroon" title="c27">n1</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="c21">u</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="c21">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n1</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">b1</font>, <font color="Maroon" title="c30">b2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1<span class="kw"> such that </span><br/><a NAME="E5:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E60">A84</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c29">b1</font>,<font color="Maroon" title="c30">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E6:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E61">A85</font></span>: 
<font color="Maroon" title="c21">u</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">Px</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c29">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c30">b2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_2_1_1_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a></span>;<br/><a NAME="E7:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E62">A86</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</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="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:37_2_1_1_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E8:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A87</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c29">b1</font>,<font color="Maroon" title="c30">b2</font></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> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b19</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c28">b19</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E5:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E60">A84</font></span></a>, <a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>;<br/><a NAME="E9:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A88</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">xb1n</font> = <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>, <font color="Maroon" title="c32">yb2n</font> = <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> as   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></span>;<br/><a NAME="E11:37_2_1_1_1_7"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c33">b1n</font> = <font color="Maroon" title="c29">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c34">b2n</font> = <font color="Maroon" title="c30">b2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></span>;<br/><span class="lab"><font color="Green" title="E65">A89</font></span>: <font color="Maroon" title="c21">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">Px</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c29">b1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c30">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E6:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E61">A85</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">xb1n</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b1n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c30">b2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c31">xb1n</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c33">b1n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">yb2n</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c34">b2n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></span>
;<br/><a NAME="E14:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E66">A90</font></span>: 
<font color="Maroon" title="c30">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c28">b19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E63">A87</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/><a NAME="E15:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E67">A91</font></span>: 
<font color="Maroon" title="c27">n1</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E7:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E62">A86</font></span></a>, <a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>;<br/><a NAME="E16:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A92</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E64">A88</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E17:37_2_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c29">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E67">A91</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E18:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A93</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><a NAME="E19:37_2_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c33">b1n</font> <a href="pre_poly.html#R3" title="PRE_POLY:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></span>;<br/><a NAME="E20:37_2_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c33">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E22:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E70">A94</font></span>: 
<font color="Maroon" title="c35">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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E23:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E71">A95</font></span>: 
<font color="Maroon" title="c33">b1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c36">i</font> = <font color="Maroon" title="c35">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a></span>;<br/><a NAME="E25:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E72">A96</font></span>: 
<font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E69">A93</font></span></a>, <a class="ref" href="pre_poly.html#D11" title="PRE_POLY:def.11">PRE_POLY:def 11</a></span>;<br/><a NAME="E26:37_2_1_1_1_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A97</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E72">A96</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
<br/>.= 
<font color="Maroon" title="c30">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E64">A88</font></span></a>, <a class="txt" href="#E14:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="ref" href="pre_poly.html#D6" title="PRE_POLY:def.6">PRE_POLY:def 6</a></span>
;<br/><span class="lab"><font color="Green" title="E74">A98</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T37" title="NAT_D:th.37">NAT_D:37</a></span>
<br/>.= 
<font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>

;<br/><a NAME="E29:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A99</font></span>: 
<font color="Maroon" title="c31">xb1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 ;<br/><a NAME="E30:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E76">A100</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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="c23">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E16:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E23:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E71">A95</font></span></a></span>;<br/><a NAME="E31:37_2_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c37">sb1n1</font> = <font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <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 <font color="Maroon" title="c1">R</font> ;<br/><a NAME="E33:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E77">A101</font></span>: 
<font color="Maroon" title="c36">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"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E16:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E23:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E71">A95</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c38">B1</font>, <font color="Maroon" title="c39">B2</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font><span class="kw"> such that </span><br/><a NAME="E35:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E78">A102</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c38">B1</font>,<font color="Maroon" title="c39">B2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E36:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E79">A103</font></span>: 
<font color="Maroon" title="c37">sb1n1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">xb1n</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c38">B1</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">yb2n</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c39">B2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E30:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E76">A100</font></span></a>, <a class="txt" href="#E28:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E74">A98</font></span></a>, <a class="txt" href="#E27:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E73">A97</font></span></a></span>;<br/><a NAME="E37:37_2_1_1_1_7"/>
<font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K11" title="PRE_POLY:func.11">Card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c36">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:37_2_1_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="txt" href="#E3:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E59">A83</font></span></a>, <a class="txt" href="#E16:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E68">A92</font></span></a>, <a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E23:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E71">A95</font></span></a></span>;<br/><a NAME="E38:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A104</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E76">A100</font></span></a>, <a class="txt" href="#E33:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E77">A101</font></span></a>, <a class="ref" href="pre_poly.html#T30" title="PRE_POLY:th.30">PRE_POLY:30</a></span>;<br/><a NAME="E39:37_2_1_1_1_7"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font> is    <a href="pre_poly.html#M2" title="PRE_POLY:mode.2">Element</a> of  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">Bags</a> <font color="Maroon" title="c2">n</font>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c40">B19</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c36">i</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c2">n</font> ;<br/><a NAME="E41:37_2_1_1_1_7"/><span class="lab"><font color="Green" title="E81">A105</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> <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="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>;<br/><a NAME="E42:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A106</font></span>: 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c38">B1</font>,<font color="Maroon" title="c39">B2</font></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> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c40">B19</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">bn</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c40">B19</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E35:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E78">A102</font></span></a>, <a class="ref" href="pre_poly.html#D17" title="PRE_POLY:def.17">PRE_POLY:def 17</a></span>;<br/><a NAME="E43:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A107</font></span>: 
<font color="Maroon" title="c38">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">B19</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/><a NAME="E44:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A108</font></span>: 
<font color="Maroon" title="c38">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">b1n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E23:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E71">A95</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>;<br/><a NAME="E45:37_2_1_1_1_7"/>
<font color="Maroon" title="c32">yb2n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c29">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E73">A97</font></span></a></span>;<br/><a NAME="E46:37_2_1_1_1_7"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c37">sb1n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37_2_1_1_1"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E18:37_2_1_1_1"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E30:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E76">A100</font></span></a>, <a class="txt" href="#E29:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E75">A99</font></span></a></span>;<br/><a NAME="E47:37_2_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A109</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c37">sb1n1</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="pre_poly.html#K21" title="PRE_POLY:func.21">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E81">A105</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E48:37_2_1_1_1_7"/>
<font color="Maroon" title="c39">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">bn</font> <a href="pre_poly.html#K13" title="PRE_POLY:func.13">-'</a> <font color="Maroon" title="c40">B19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E82">A106</font></span></a>, <a class="ref" href="finseq_1.html#T98" title="FINSEQ_1:th.98">FINSEQ_1:98</a></span>;<br/><a NAME="E49:37_2_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c39">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">b2n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E64">A88</font></span></a>, <a class="txt" href="#E14:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="txt" href="#E43:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E83">A107</font></span></a>, <a class="txt" href="#E44:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E84">A108</font></span></a>, <a class="ref" href="hilbasis.html#T5" target="_self" title="HILBASIS:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E50:37_2_1_1_1_7"/>
<font color="Maroon" title="c21">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">p</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c27">n1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E65">A89</font></span></a>, <a class="txt" href="#E22:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E70">A94</font></span></a>, <a class="txt" href="#E38:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E80">A104</font></span></a>, <a class="txt" href="#E36:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E79">A103</font></span></a>, <a class="txt" href="#E44:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E84">A108</font></span></a>, <a class="txt" href="#E47:37_2_1_1_1_7"><span class="lab"><font color="Green" title="E85">A109</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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><a NAME="E46:37_2_1_1_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</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"><font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/><a NAME="E47:37_2_1_1_1"/><span class="kw">then </span>
 <a href="matrlin.html#K6" title="MATRLIN:func.6">Sum</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:37_2_1_1_1"><span class="lab"><font color="Green" title="E56">A78</font></span></a>, <a class="txt" href="#E44:37_2_1_1_1"><span class="lab"><font color="Green" title="E57">A79</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>;<br/><a NAME="E48:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A110</font></span>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Pxy</font> <a href="polynom1.html#K4" title="POLYNOM1:func.4">.</a> <font color="Maroon" title="c17">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:37_2_1_1_1"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="polynom1.html#T34" title="POLYNOM1:th.34">POLYNOM1:34</a></span>;<br/><span class="lab"><font color="Green" title="E60">A111</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c21">u</font> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T104" title="CARD_1:th.104">CARD_1:104</a></span>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c26">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:37_2_1_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="card_1.html#T104" title="CARD_1:th.104">CARD_1:104</a></span>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:37_2_1_1_1"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="txt" href="#E41:37_2_1_1_1"><span class="lab"><font color="Green" title="E55">A77</font></span></a>, <a class="ref" href="pre_poly.html#T19" title="PRE_POLY:th.19">PRE_POLY:19</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T104" title="CARD_1:th.104">CARD_1:104</a></span>
;<br/><a NAME="E50:37_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A112</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</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="pre_poly.html#K5" title="PRE_POLY:func.5">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/><a NAME="E51:37_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c26">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c21">u</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:37_2_1_1_1"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="txt" href="#E41:37_2_1_1_1"><span class="lab"><font color="Green" title="E55">A77</font></span></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></span>;<br/><span class="kw">hence </span><a NAME="E52:37_2_1_1_1"/>
<font color="Maroon" title="c12">Pxy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">PxPy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:37_2_1_1_1"><span class="lab"><font color="Green" title="E59">A110</font></span></a>, <a class="txt" href="#E11:37_2_1_1_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E45:37_2_1_1_1"><span class="lab"><font color="Green" title="E58">A82</font></span></a>, <a class="txt" href="#E49:37_2_1_1_1"><span class="lab"><font color="Green" title="E60">A111</font></span></a>, <a class="txt" href="#E50:37_2_1_1_1"><span class="lab"><font color="Green" title="E61">A112</font></span></a>, <a class="ref" href="rlvect_2.html#T8" title="RLVECT_2:th.8">RLVECT_2:8</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>
<span class="kw">hence </span><a NAME="E6:37_2_1_1"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x9</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c8">y9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x9</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K6" title="HILBASIS:func.6">upm</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</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>
