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<b>set </b><font color="Maroon" title="c3">PNR</font> =  <a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">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/>
<b>set </b><font color="Maroon" title="c6">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/>
<b>thus </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x'</font>, <font color="Maroon" title="c8">y'</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#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">x'</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y'</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">x'</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">y'</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x'</font>, <font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c8">y'</font>, <font color="Maroon" title="c11">xy</font> = <font color="Maroon" title="c7">x'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">y'</font> as   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><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">x'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">y'</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">x'</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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">y'</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">x'</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">y'</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">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> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E3:37_2_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
<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#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c15">Py</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E4:37_2_1_1"/><i><font color="Green" title="E34">A5</font></i>: 
<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#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c10">y</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">b'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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">b'</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">b'</font> )</span><br/><b>assume </b><a NAME="E1:37_2_1_1_1"/>
<font color="Maroon" title="c16">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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:()"><i><font color="Red">::  thesis: </font></i></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">b'</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">b'</font></span><br/><b>then reconsider </b><font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c16">b'</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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/><b>consider </b><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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:37_2_1_1_1"/><i><font color="Green" title="E35">A6</font></i>: 
(  <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 &amp; <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> &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="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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="polynom3.html#D11" title="POLYNOM3:def.11">POLYNOM3:def 11</a></i>;<br/><a NAME="E5:37_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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><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="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></i>;<br/><b>reconsider </b><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#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>consider </b><font color="Maroon" title="c21">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#K29" title="POLYNOM1:func.29">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><b> such that </b><br/><a NAME="E9:37_2_1_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
( ( for <font color="Olive" title="b1">p</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> holds  <font color="Maroon" title="c21">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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c19">bn</font> ) &amp; <font color="Maroon" title="c20">xybn</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</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="c21">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="hilbasis.html#T29" target="_self" title="HILBASIS:th.29">Th29</a></i>;<br/><a NAME="E10:37_2_1_1_1"/><i><font color="Green" title="E37">A8</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>;<br/><b>defpred </b><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> ] <b>means </b>for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</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#K28" title="POLYNOM1:func.28">*'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</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="polynom1.html#K20" title="POLYNOM1:func.20">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">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#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b6">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b7">b2</font></span>)</span> ) ) );<br/><div><i><font color="Green" title="E38">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">e'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">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> 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">e'</font>,<font color="Olive" title="b1">u</font>] ) )</span><br/><b>assume </b><a NAME="E1:37_2_1_1_1_1"/>
<font color="Maroon" title="c22">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">e'</font>,<font color="Olive" title="b1">u</font>] )</span><br/><b>then reconsider </b><font color="Maroon" title="c23">e</font> = <font color="Maroon" title="c22">e'</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/><b>reconsider </b><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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c23">e</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>consider </b><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><b> such that </b><br/><a NAME="E5:37_2_1_1_1_1"/><i><font color="Green" title="E39">A10</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c24">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c25">q</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</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="polynom1.html#K20" title="POLYNOM1:func.20">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">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#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">q</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/><a NAME="E6:37_2_1_1_1_1"/><i><font color="Green" title="E40">A11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c23">e</font>,<font color="Maroon" title="c26">fcr</font>]
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:37_2_1_1_1_1"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/><a NAME="E7:37_2_1_1_1_1"/>
<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> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/><b>hence </b><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">e'</font>,<font color="Olive" title="b1">u</font>] )
 <b>by </b><i><a class="txt" href="hilbasis.html#E6:37_2_1_1_1_1"><i><font color="Green" title="E40">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c22">s</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c18">r</font>,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> <b> such that </b><br/><a NAME="E13:37_2_1_1_1"/><i><font color="Green" title="E39">A12</font></i>: 
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>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="hilbasis.html#E11:37_2_1_1_1"><i><font color="Green" title="E38">A9</font></i></a>);<br/></i><a NAME="E14:37_2_1_1_1"/><i><font color="Green" title="E40">A13</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E15:37_2_1_1_1"/><b>then </b>
 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#K4" title="RELSET_1:func.4">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>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E16:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">A14</font></i>: 
<font color="Maroon" title="c22">s</font> is <a href="finseq_1.html#V1" title="FINSEQ_1:attr.1">FinSequence-like</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E17:37_2_1_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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/><b>then reconsider </b><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>  <b>by </b><i><a class="txt" href="hilbasis.html#E16:37_2_1_1_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><a NAME="E19:37_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#K8" title="MATRLIN:func.8">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>
 <b>by </b><i><a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></i>;<br/><a NAME="E20:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E42">A15</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E21:37_2_1_1_1"/><i><font color="Green" title="E43">A16</font></i>: 
 <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="c21">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">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#K8" title="MATRLIN:func.8">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="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="c24">k</font> )</span><br/><b>assume </b><a NAME="E1:37_2_1_1_1_2"/><i><font color="Green" title="E44">A17</font></i>: 
<font color="Maroon" title="c24">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">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="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">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="c24">k</font></span><br/><a NAME="E2:37_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E45">A18</font></i>: 
<span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">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="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">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="c24">k</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E3:37_2_1_1_1_2"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c23">s</font> <a href="matrlin.html#K1" title="MATRLIN:func.1">/.</a> <font color="Maroon" title="c24">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="c24">k</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">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="c24">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>  <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">sk</font> = <font color="Maroon" title="c25">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/><b>reconsider </b><font color="Maroon" title="c27">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="c24">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>, <font color="Maroon" title="c28">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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c24">k</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c29">pq'</font> = <font color="Maroon" title="c27">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c28">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#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><i><font color="Green" title="E47">A20</font></i>: <font color="Maroon" title="c18">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">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="c24">k</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c24">k</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c29">pq'</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="matrlin.html#K8" title="MATRLIN:func.8">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="c24">k</font> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c26">sk</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="hilbasis.html#E2:37_2_1_1_1_2"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="hilbasis.html#E3:37_2_1_1_1_2"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="matrlin.html#D8" title="MATRLIN:def.8">MATRLIN:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">p</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c28">q</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c19">bn</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">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="c24">k</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E9:37_2_1_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="txt" href="hilbasis.html#E8:37_2_1_1_1_2"><i><font color="Green" title="E47">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">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="c24">k</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_2"><i><font color="Green" title="E44">A17</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E23:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E44">A21</font></i>: 
 <a href="matrlin.html#K8" title="MATRLIN:func.8">Sum</a> <font color="Maroon" title="c23">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">g</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c18">r</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="hilbasis.html#E21:37_2_1_1_1"><i><font color="Green" title="E43">A16</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/><b>set </b><font color="Maroon" title="c24">t</font> =  <a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font>;<br/><a NAME="E24:37_2_1_1_1"/><i><font color="Green" title="E45">A22</font></i>: 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:37_2_1_1_1"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="hilbasis.html#E23:37_2_1_1_1"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="polynom1.html#T34" title="POLYNOM1:th.34">POLYNOM1:34</a></i>;<br/><b>consider </b><font color="Maroon" title="c25">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><b> such that </b><br/><a NAME="E26:37_2_1_1_1"/><i><font color="Green" title="E46">A23</font></i>: 
( <font color="Maroon" title="c13">PxPy</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</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="c25">u</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</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="c25">u</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">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="polynom1.html#K20" title="POLYNOM1:func.20">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="c25">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#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:37_2_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="polynom1.html#D26" title="POLYNOM1:def.26">POLYNOM1:def 26</a></i>;<br/><b>defpred </b><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> ] <b>means </b>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="polynom1.html#NM1" title="POLYNOM1:NM.1">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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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/><div><i><font color="Green" title="E47">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c26">n1'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">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="c25">u</font> implies  ex <font color="Olive" title="b1">n2'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n2'</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="dtconstr.html#K15" title="DTCONSTR:func.15">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">n1'</font>,<font color="Olive" title="b1">n2'</font>] ) )</span><br/><b>assume </b><a NAME="E1:37_2_1_1_1_3"/><i><font color="Green" title="E48">A25</font></i>: 
<font color="Maroon" title="c26">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="c25">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n2'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n2'</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="dtconstr.html#K15" title="DTCONSTR:func.15">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">n1'</font>,<font color="Olive" title="b1">n2'</font>] )</span><br/><b>then reconsider </b><font color="Maroon" title="c27">n1</font> = <font color="Maroon" title="c26">n1'</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="c25">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E4:37_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E49">A26</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_3"><i><font color="Green" title="E48">A25</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c28">b1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E6:37_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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><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="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></i>;<br/><a NAME="E8:37_2_1_1_1_3"/><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c28">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_3"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><a NAME="E9:37_2_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c29">b1n</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></i>;<br/><a NAME="E10:37_2_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c29">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c30">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:37_2_1_1_1_3"/><i><font color="Green" title="E51">A28</font></i>: 
( <font color="Maroon" title="c30">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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <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="polynom1.html#K19" title="POLYNOM1:func.19">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">i</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c31">i</font> = <font color="Maroon" title="c30">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>  <b>by </b><i><a class="txt" href="hilbasis.html#E12:37_2_1_1_1_3"><i><font color="Green" title="E51">A28</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c32">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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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="c31">i</font>;<br/><a NAME="E14:37_2_1_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:37_2_1_1_1_3"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/><a NAME="E15:37_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E52">A29</font></i>: 
<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E16:37_2_1_1_1_3"/><i><font color="Green" title="E53">A30</font></i>: 
<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> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E17:37_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E54">A31</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E15:37_2_1_1_1_3"><i><font color="Green" title="E52">A29</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>, <font color="Maroon" title="c34">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="binarith.html#K3" title="BINARITH:func.3">-'</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#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E19:37_2_1_1_1_3"/>
<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> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E17:37_2_1_1_1_3"><i><font color="Green" title="E54">A31</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E20:37_2_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c34">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="binarith.html#K3" title="BINARITH:func.3">-'</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> &amp; <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_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="E21:37_2_1_1_1_3"/><b>then </b>
 <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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E17:37_2_1_1_1_3"><i><font color="Green" title="E54">A31</font></i></a></i>;<br/><b>then </b><i><font color="Green" title="E55">A32</font></i>:  <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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E23:37_2_1_1_1_3"/><b>then </b><i><font color="Green" title="E56">A33</font></i>: 
<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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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="c31">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:37_2_1_1_1_3"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="hilbasis.html#E17:37_2_1_1_1_3"><i><font color="Green" title="E54">A31</font></i></a>, <a class="ref" href="polynom1.html#T33" title="POLYNOM1:th.33">POLYNOM1:33</a></i>;<br/><a NAME="E24:37_2_1_1_1_3"/>
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="polynom1.html#NM1" title="POLYNOM1:NM.1">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="c27">n1</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="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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; <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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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="c31">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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E12:37_2_1_1_1_3"><i><font color="Green" title="E51">A28</font></i></a>, <a class="txt" href="hilbasis.html#E15:37_2_1_1_1_3"><i><font color="Green" title="E52">A29</font></i></a>, <a class="txt" href="hilbasis.html#E16:37_2_1_1_1_3"><i><font color="Green" title="E53">A30</font></i></a>, <a class="txt" href="hilbasis.html#E22:37_2_1_1_1_3"><i><font color="Green" title="E55">A32</font></i></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></i>;<br/><b>hence </b><a NAME="E25:37_2_1_1_1_3"/>
 ex <font color="Olive" title="b1">n2'</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">n2'</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="dtconstr.html#K15" title="DTCONSTR:func.15">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">n1'</font>,<font color="Olive" title="b1">n2'</font>] )
 <b>by </b><i><a class="txt" href="hilbasis.html#E23:37_2_1_1_1_3"><i><font color="Green" title="E56">A33</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c26">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">u</font>, <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span><b> such that </b><br/><a NAME="E29:37_2_1_1_1"/><i><font color="Green" title="E48">A34</font></i>: 
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="c25">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>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="hilbasis.html#E27:37_2_1_1_1"><i><font color="Green" title="E47">A24</font></i></a>);<br/></i><div><i><font color="Green" title="E49">A35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_4"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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="c25">u</font> implies <font color="Maroon" title="c25">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/><b>assume </b><a NAME="E1:37_2_1_1_1_4"/><i><font color="Green" title="E50">A36</font></i>: 
<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="c25">u</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c25">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="dtconstr.html#K15" title="DTCONSTR:func.15">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/><b>then consider </b><font color="Maroon" title="c28">b1</font>, <font color="Maroon" title="c29">b2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1<b> such that </b><br/><a NAME="E3:37_2_1_1_1_4"/><i><font color="Green" title="E51">A37</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">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="c28">b1</font>,<font color="Maroon" title="c29">b2</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> &amp; <font color="Maroon" title="c25">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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c28">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c29">b2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c30">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="c28">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="c31">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="c29">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#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E5:37_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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">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>, <font color="Maroon" title="c33">b2n</font> = <font color="Maroon" title="c29">b2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></i>;<br/><i><font color="Green" title="E52">A38</font></i>: <font color="Maroon" title="c25">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#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c28">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c29">b2</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_4"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="hilbasis.html#E3:37_2_1_1_1_4"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">xb1n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c32">b1n</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">Py</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c29">b2</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">xb1n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c32">b1n</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">yb2n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c33">b2n</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#D6" target="_self" title="HILBASIS:def.6">Def6</a></i>
;<br/><a NAME="E8:37_2_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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/><b>then reconsider </b><font color="Maroon" title="c34">b1'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ;<br/><a NAME="E10:37_2_1_1_1_4"/><i><font color="Green" title="E53">A39</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E11:37_2_1_1_1_4"/><b>then </b>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c28">b1</font>,<font color="Maroon" title="c29">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="c34">b1'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c34">b1'</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_4"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="hilbasis.html#E3:37_2_1_1_1_4"><i><font color="Green" title="E51">A37</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/><a NAME="E12:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E54">A40</font></i>: 
( <font color="Maroon" title="c28">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">b1'</font> &amp; <font color="Maroon" title="c29">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">b</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c34">b1'</font> )
 <b>by </b><i><a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/><a NAME="E13:37_2_1_1_1_4"/><i><font color="Green" title="E55">A41</font></i>: 
<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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_4"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="hilbasis.html#E10:37_2_1_1_1_4"><i><font color="Green" title="E53">A39</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/><a NAME="E14:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E56">A42</font></i>: 
<font color="Maroon" title="c28">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:37_2_1_1_1_4"><i><font color="Green" title="E54">A40</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E15:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c28">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1_4"><i><font color="Green" title="E55">A41</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E16:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E57">A43</font></i>: 
<font color="Maroon" title="c28">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><a NAME="E17:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c32">b1n</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></i>;<br/><a NAME="E18:37_2_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c32">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E20:37_2_1_1_1_4"/><i><font color="Green" title="E58">A44</font></i>: 
( <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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <font color="Maroon" title="c32">b1n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><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>  <b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1_4"><i><font color="Green" title="E58">A44</font></i></a></i>;<br/><a NAME="E22:37_2_1_1_1_4"/><i><font color="Green" title="E59">A45</font></i>: 
( <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> &amp; <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="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> &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">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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E29:37_2_1_1_1"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_4"><i><font color="Green" title="E50">A36</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1_4"><i><font color="Green" title="E56">A42</font></i></a>, <a class="txt" href="hilbasis.html#E20:37_2_1_1_1_4"><i><font color="Green" title="E58">A44</font></i></a></i>;<br/><a NAME="E23:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E60">A46</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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="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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">i</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T33" title="POLYNOM1:th.33">POLYNOM1:33</a></i>;<br/><i><font color="Green" title="E61">A47</font></i>: <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> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 = 
<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> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<b>by </b><i><a class="ref" href="binarith.html#T55" title="BINARITH:th.55">BINARITH:55</a></i>
<br/>.= 
<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>

;<br/><a NAME="E25:37_2_1_1_1_4"/><i><font color="Green" title="E62">A48</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E16:37_2_1_1_1_4"><i><font color="Green" title="E57">A43</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/><a NAME="E26:37_2_1_1_1_4"/><b>then </b>
<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A49</font></i>: <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="binarith.html#K3" title="BINARITH:func.3">-'</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> = 
<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="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>
<b>by </b><i><a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<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="c28">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="binarith.html#K3" title="BINARITH:func.3">-'</a> <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>
<b>by </b><i><a class="txt" href="hilbasis.html#E25:37_2_1_1_1_4"><i><font color="Green" title="E62">A48</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon" title="c29">b2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E12:37_2_1_1_1_4"><i><font color="Green" title="E54">A40</font></i></a>, <a class="ref" href="polynom1.html#D6" title="POLYNOM1:def.6">POLYNOM1:def 6</a></i>
;<br/><a NAME="E28:37_2_1_1_1_4"/>
<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> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E22:37_2_1_1_1_4"><i><font color="Green" title="E59">A45</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><b>then reconsider </b><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="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> 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="E30:37_2_1_1_1_4"/>
( <font color="Maroon" title="c30">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="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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c31">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="binarith.html#K3" title="BINARITH:func.3">-'</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> &amp; <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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c37">sb1n1</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E24:37_2_1_1_1_4"><i><font color="Green" title="E61">A47</font></i></a>, <a class="txt" href="hilbasis.html#E27:37_2_1_1_1_4"><i><font color="Green" title="E63">A49</font></i></a></i>;<br/><a NAME="E31:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E64">A50</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c30">xb1n</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c31">yb2n</font></span>)</span> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</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="c37">sb1n1</font> &amp;  <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="polynom1.html#K20" title="POLYNOM1:func.20">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="c37">sb1n1</font> holds <br/> ex <font color="Olive" title="b2">b1</font>, <font color="Olive" title="b3">b2</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">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="c37">sb1n1</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="c30">xb1n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">yb2n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b3">b2</font></span>)</span> ) ) )
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E22:37_2_1_1_1_4"><i><font color="Green" title="E59">A45</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c38">B1</font>, <font color="Maroon" title="c39">B2</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font><b> such that </b><br/><a NAME="E33:37_2_1_1_1_4"/><i><font color="Green" title="E65">A51</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">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> &amp; <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="c30">xb1n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c38">B1</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">yb2n</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c39">B2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E22:37_2_1_1_1_4"><i><font color="Green" title="E59">A45</font></i></a>, <a class="txt" href="hilbasis.html#E24:37_2_1_1_1_4"><i><font color="Green" title="E61">A47</font></i></a>, <a class="txt" href="hilbasis.html#E27:37_2_1_1_1_4"><i><font color="Green" title="E63">A49</font></i></a></i>;<br/><a NAME="E34:37_2_1_1_1_4"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c40">B1'</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> ;<br/><a NAME="E36:37_2_1_1_1_4"/><i><font color="Green" title="E66">A52</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/><a NAME="E37:37_2_1_1_1_4"/><b>then </b>
<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">B1'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">bn</font> <a href="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c40">B1'</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1_4"><i><font color="Green" title="E58">A44</font></i></a>, <a class="txt" href="hilbasis.html#E33:37_2_1_1_1_4"><i><font color="Green" title="E65">A51</font></i></a>, <a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>;<br/><a NAME="E38:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E67">A53</font></i>: 
( <font color="Maroon" title="c38">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c40">B1'</font> &amp; <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="polynom1.html#K9" title="POLYNOM1:func.9">-'</a> <font color="Maroon" title="c40">B1'</font> )
 <b>by </b><i><a class="ref" href="group_7.html#T2" title="GROUP_7:th.2">GROUP_7:2</a></i>;<br/><a NAME="E39:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E68">A54</font></i>: 
<font color="Maroon" title="c38">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">b1n</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E20:37_2_1_1_1_4"><i><font color="Green" title="E58">A44</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E40:37_2_1_1_1_4"/><b>then </b><i><font color="Green" title="E69">A55</font></i>: 
<font color="Maroon" title="c39">B2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c33">b2n</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E12:37_2_1_1_1_4"><i><font color="Green" title="E54">A40</font></i></a>, <a class="txt" href="hilbasis.html#E38:37_2_1_1_1_4"><i><font color="Green" title="E67">A53</font></i></a>, <a class="ref" href="hilbasis.html#T5" target="_self" title="HILBASIS:th.5">Th5</a></i>;<br/><a NAME="E41:37_2_1_1_1_4"/>
 <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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E31:37_2_1_1_1_4"><i><font color="Green" title="E64">A50</font></i></a>, <a class="txt" href="hilbasis.html#E36:37_2_1_1_1_4"><i><font color="Green" title="E66">A52</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><b>hence </b><a NAME="E42:37_2_1_1_1_4"/>
<font color="Maroon" title="c25">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="dtconstr.html#K15" title="DTCONSTR:func.15">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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:37_2_1_1_1_4"><i><font color="Green" title="E52">A38</font></i></a>, <a class="txt" href="hilbasis.html#E20:37_2_1_1_1_4"><i><font color="Green" title="E58">A44</font></i></a>, <a class="txt" href="hilbasis.html#E23:37_2_1_1_1_4"><i><font color="Green" title="E60">A46</font></i></a>, <a class="txt" href="hilbasis.html#E33:37_2_1_1_1_4"><i><font color="Green" title="E65">A51</font></i></a>, <a class="txt" href="hilbasis.html#E39:37_2_1_1_1_4"><i><font color="Green" title="E68">A54</font></i></a>, <a class="txt" href="hilbasis.html#E40:37_2_1_1_1_4"><i><font color="Green" title="E69">A55</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E31:37_2_1_1_1"/><i><font color="Green" title="E50">A56</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span> <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="E32:37_2_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <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> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E33:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E51">A57</font></i>: 
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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E34:37_2_1_1_1"/><b>then </b>
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="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/><a NAME="E35:37_2_1_1_1"/><b>then </b>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1
 <b>by </b><i><a class="ref" href="polynom3.html#T4" title="POLYNOM3:th.4">POLYNOM3:4</a></i>;<br/><a NAME="E36:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E52">A58</font></i>: 
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">Card</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E33:37_2_1_1_1"><i><font color="Green" title="E51">A57</font></i></a>, <a class="ref" href="card_3.html#D2" title="CARD_3:def.2">CARD_3:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_5"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c27">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">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>, <font color="Maroon" title="c28">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E2:37_2_1_1_1_5"/>
<font color="Maroon" title="c23">s</font> <a href="matrlin.html#K1" title="MATRLIN: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>
 ;<br/><a NAME="E3:37_2_1_1_1_5"/><b>then </b><i><font color="Green" title="E53">A59</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E33:37_2_1_1_1"><i><font color="Green" title="E51">A57</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:37_2_1_1_1_5"/>
( <font color="Maroon" title="c27">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">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c28">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 ;<br/><a NAME="E5:37_2_1_1_1_5"/><b>then </b>
 <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#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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E33:37_2_1_1_1"><i><font color="Green" title="E51">A57</font></i></a>, <a class="txt" href="hilbasis.html#E3:37_2_1_1_1_5"><i><font color="Green" title="E53">A59</font></i></a></i>;<br/><b>hence </b><a NAME="E6:37_2_1_1_1_5"/>
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E38:37_2_1_1_1"/><b>then </b>
 <a href="wsierp_1.html#K3" title="WSIERP_1:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E36:37_2_1_1_1"><i><font color="Green" title="E52">A58</font></i></a></i>;<br/><a NAME="E39:37_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">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> 
 <b>by </b><i><a class="ref" href="polynom1.html#T30" title="POLYNOM1:th.30">POLYNOM1:30</a></i>;<br/><a NAME="E40:37_2_1_1_1"/><b>then </b>
 <a href="dtconstr.html#K15" title="DTCONSTR:func.15">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> 
 <b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/><a NAME="E41:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E53">A60</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">u</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>  &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="hilbasis.html#E31:37_2_1_1_1"><i><font color="Green" title="E50">A56</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c27">n1'</font>, <font color="Maroon" title="c28">n2'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">p</font> &amp; <font color="Maroon" title="c28">n2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">n1'</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">n2'</font> implies <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n2'</font> )</span><br/><b>assume </b><a NAME="E1:37_2_1_1_1_6"/><i><font color="Green" title="E54">A61</font></i>: 
( <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c26">p</font> &amp; <font color="Maroon" title="c28">n2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">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">n1'</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">n2'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n2'</font></span><br/><a NAME="E2:37_2_1_1_1_6"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c25">u</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E41:37_2_1_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">n1</font> = <font color="Maroon" title="c27">n1'</font>, <font color="Maroon" title="c30">n2</font> = <font color="Maroon" title="c28">n2'</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>  <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a></i>;<br/><i><font color="Green" title="E55">A62</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
;<br/><a NAME="E5:37_2_1_1_1_6"/><b>then </b><i><font color="Green" title="E56">A63</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c31">b1</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E7:37_2_1_1_1_6"/><i><font color="Green" title="E57">A64</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_6"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c32">b2</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E9:37_2_1_1_1_6"/>
<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
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><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="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self" title="HILBASIS:th.3">Th3</a></i>;<br/><a NAME="E11:37_2_1_1_1_6"/>
<font color="Maroon" title="c31">b1</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E5:37_2_1_1_1_6"><i><font color="Green" title="E56">A63</font></i></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><a NAME="E12:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c33">b1n</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></i>;<br/><a NAME="E13:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c33">b1n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c35">i1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:37_2_1_1_1_6"/><i><font color="Green" title="E58">A65</font></i>: 
( <font color="Maroon" title="c35">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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <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="polynom1.html#K19" title="POLYNOM1:func.19">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">i1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c36">i1</font> = <font color="Maroon" title="c35">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>  <b>by </b><i><a class="txt" href="hilbasis.html#E15:37_2_1_1_1_6"><i><font color="Green" title="E58">A65</font></i></a></i>;<br/><a NAME="E17:37_2_1_1_1_6"/>
<font color="Maroon" title="c32">b2</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:37_2_1_1_1_6"><i><font color="Green" title="E57">A64</font></i></a>, <a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><a NAME="E18:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c34">b2n</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T4" target="_self" title="HILBASIS:th.4">Th4</a></i>;<br/><a NAME="E19:37_2_1_1_1_6"/><b>then </b>
<font color="Maroon" title="c34">b2n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c37">i2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E21:37_2_1_1_1_6"/><i><font color="Green" title="E59">A66</font></i>: 
( <font color="Maroon" title="c37">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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <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="polynom1.html#K19" title="POLYNOM1:func.19">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">i2</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c38">i2</font> = <font color="Maroon" title="c37">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>  <b>by </b><i><a class="txt" href="hilbasis.html#E21:37_2_1_1_1_6"><i><font color="Green" title="E59">A66</font></i></a></i>;<br/><a NAME="E23:37_2_1_1_1_6"/><i><font color="Green" title="E60">A67</font></i>: 
( <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="c36">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> &amp; <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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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">i1</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E29:37_2_1_1_1"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="hilbasis.html#E15:37_2_1_1_1_6"><i><font color="Green" title="E58">A65</font></i></a></i>;<br/><a NAME="E24:37_2_1_1_1_6"/><i><font color="Green" title="E61">A68</font></i>: 
( <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="c38">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> &amp; <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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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">i2</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E29:37_2_1_1_1"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="hilbasis.html#E21:37_2_1_1_1_6"><i><font color="Green" title="E59">A66</font></i></a></i>;<br/><a NAME="E25:37_2_1_1_1_6"/>
( <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom3.html#T16" title="POLYNOM3:th.16">POLYNOM3:16</a></i>;<br/><a NAME="E26:37_2_1_1_1_6"/><b>then </b><i><font color="Green" title="E62">A69</font></i>: 
( <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="c36">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c38">i2</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="hilbasis.html#E23:37_2_1_1_1_6"><i><font color="Green" title="E60">A67</font></i></a>, <a class="txt" href="hilbasis.html#E24:37_2_1_1_1_6"><i><font color="Green" title="E61">A68</font></i></a>, <a class="ref" href="polynom3.html#T22" title="POLYNOM3:th.22">POLYNOM3:22</a></i>;<br/><a NAME="E27:37_2_1_1_1_6"/><i><font color="Green" title="E63">A70</font></i>: 
( <font color="Maroon" title="c31">b1</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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> &amp; <font color="Maroon" title="c32">b2</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">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> )
 <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>then </b><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>
<b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c32">b2</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E15:37_2_1_1_1_6"><i><font color="Green" title="E58">A65</font></i></a>, <a class="txt" href="hilbasis.html#E21:37_2_1_1_1_6"><i><font color="Green" title="E59">A66</font></i></a>, <a class="txt" href="hilbasis.html#E26:37_2_1_1_1_6"><i><font color="Green" title="E62">A69</font></i></a>, <a class="txt" href="hilbasis.html#E27:37_2_1_1_1_6"><i><font color="Green" title="E63">A70</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E29:37_2_1_1_1_6"/>
<font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">n2'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_6"><i><font color="Green" title="E54">A61</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_6"><i><font color="Green" title="E55">A62</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E43:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E54">A71</font></i>: 
<font color="Maroon" title="c26">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E44:37_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="dtconstr.html#K15" title="DTCONSTR:func.15">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#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">n1'</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span> or <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">p</font> )</span><br/>

<b>assume </b><a NAME="E1:37_2_1_1_1_7"/><i><font color="Green" title="E55">A72</font></i>: 
<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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">p</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c28">n1</font> = <font color="Maroon" title="c27">n1'</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/>
<b>consider </b><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> <b> such that </b><br/><a NAME="E4:37_2_1_1_1_7"/><i><font color="Green" title="E56">A73</font></i>: 
( <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> &amp; <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> &amp; <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="polynom1.html#K5" title="POLYNOM1:func.5">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="binarith.html#K3" title="BINARITH:func.3">-'</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> &amp; <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="dtconstr.html#K15" title="DTCONSTR:func.15">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> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_7"><i><font color="Green" title="E55">A72</font></i></a>, <a class="ref" href="polynom1.html#T32" title="POLYNOM1:th.32">POLYNOM1:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c31">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="c29">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>, <font color="Maroon" title="c32">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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c29">i</font></span>)</span> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c33">bj</font> = <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E7:37_2_1_1_1_7"/>
<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> 
 
<b>by </b><i><a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E8:37_2_1_1_1_7"/><b>then </b>
( <font color="Maroon" title="c29">i</font> is    <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>  &amp; <font color="Maroon" title="c31">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="Maroon" title="c29">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c32">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="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c29">i</font></span>)</span> &amp; <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> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
<a NAME="E9:37_2_1_1_1_7"/><b>then </b>
 <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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a></i>;<br/>
<b>then </b><i><font color="Green" title="E57">A74</font></i>:  <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="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
;<br/>
<a NAME="E11:37_2_1_1_1_7"/><b>then </b><i><font color="Green" title="E58">A75</font></i>: 
<font color="Maroon" title="c33">bj</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E12:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c33">bj</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="txt" href="hilbasis.html#E10:37_2_1_1_1_7"><i><font color="Green" title="E57">A74</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E13:37_2_1_1_1_7"/><b>then </b><i><font color="Green" title="E59">A76</font></i>: 
<font color="Maroon" title="c33">bj</font> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c19">bn</font>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c34">bij</font> = <font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_1_1_1_7_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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_7_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_7_2_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_7_2_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1"/><i><font color="Green" title="E60">A77</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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_7_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_2_1_1_1_7_2_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_7_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1_1_1_1"/><i><font color="Green" title="E61">A78</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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"><b>then </b><i><font color="Green" title="E62">A79</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c33">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c35">k</font>
<b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
;<br/><a NAME="E3:37_2_1_1_1_7_2_1_1_1_1_1"/>
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_7_2_1_1_1_1_1"><i><font color="Green" title="E61">A78</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:37_2_1_1_1_7_2_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E13:37_2_1_1_1_7"><i><font color="Green" title="E59">A76</font></i></a>, <a class="txt" href="hilbasis.html#E2:37_2_1_1_1_7_2_1_1_1_1_1"><i><font color="Green" title="E62">A79</font></i></a>, <a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_1_1_1_2"/><i><font color="Green" title="E61">A80</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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"><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>
<b>by </b><i><a class="ref" href="nat_1.html#T39" title="NAT_1:th.39">NAT_1:39</a></i>
<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>
<b>by </b><i><a class="ref" href="ordinal1.html#D1" title="ORDINAL1:def.1">ORDINAL1:def 1</a></i>
;<br/><a NAME="E3:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_7_2_1_1"><i><font color="Green" title="E60">A77</font></i></a>, <a class="txt" href="hilbasis.html#E1:37_2_1_1_1_7_2_1_1_1_1_2"><i><font color="Green" title="E61">A80</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E62">A81</font></i>: 
<font color="Maroon" title="c35">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E63">A82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>;<br/><a NAME="E6:37_2_1_1_1_7_2_1_1_1_1_2"/><i><font color="Green" title="E64">A83</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">i</font> &amp; <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 )
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:37_2_1_1_1_7_2_1_1_1_1_2"/><b>then </b>
<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
 <b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>hence </b><a NAME="E8:37_2_1_1_1_7_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7_2_1_1_1_1_2"><i><font color="Green" title="E62">A81</font></i></a>, <a class="txt" href="hilbasis.html#E5:37_2_1_1_1_7_2_1_1_1_1_2"><i><font color="Green" title="E63">A82</font></i></a>, <a class="txt" href="hilbasis.html#E6:37_2_1_1_1_7_2_1_1_1_1_2"><i><font color="Green" title="E64">A83</font></i></a>, <a class="ref" href="binarith.html#T50" title="BINARITH:th.50">BINARITH:50</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:37_2_1_1_1_7_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1_1_7_2_1_2"><b>suppose </b></a><a NAME="E1:37_2_1_1_1_7_2_1_2"/><i><font color="Green" title="E60">A84</font></i>: 
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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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_7_2_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="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><b>hence </b><a NAME="E3:37_2_1_1_1_7_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:37_2_1_1_1_7_2_1_2"><i><font color="Green" title="E60">A84</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E15:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="polynom1.html#R3" title="POLYNOM1:pred.3">divides</a> <font color="Maroon" title="c17">b</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D13" title="POLYNOM1:def.13">POLYNOM1:def 13</a></i>;<br/>
<a NAME="E16:37_2_1_1_1_7"/><b>then </b>
<font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T7" target="_self" title="HILBASIS:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c35">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E18:37_2_1_1_1_7"/><i><font color="Green" title="E60">A85</font></i>: 
( <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="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span> &amp; <font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:37_2_1_1_1_7"/><i><font color="Green" title="E61">A86</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c25">u</font>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E41:37_2_1_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<i><font color="Green" title="E62">A87</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">u</font> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K20" title="POLYNOM1:func.20">decomp</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c17">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D19" title="POLYNOM1:def.19">POLYNOM1:def 19</a></i>
;<br/>
<a NAME="E21:37_2_1_1_1_7"/>
( <font color="Maroon" title="c35">k</font> is    <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>  &amp; <font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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> &amp; <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"><a href="polynom1.html#K19" title="POLYNOM1:func.19">divisors</a> <font color="Maroon" title="c19">bn</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="polynom1.html#K19" title="POLYNOM1:func.19">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="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="txt" href="hilbasis.html#E10:37_2_1_1_1_7"><i><font color="Green" title="E57">A74</font></i></a>, <a class="txt" href="hilbasis.html#E11:37_2_1_1_1_7"><i><font color="Green" title="E58">A75</font></i></a>, <a class="txt" href="hilbasis.html#E18:37_2_1_1_1_7"><i><font color="Green" title="E60">A85</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>;<br/>
<a NAME="E22:37_2_1_1_1_7"/><b>then </b><i><font color="Green" title="E63">A88</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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#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="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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c33">bj</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <span class="p5">(<span class="default"><font color="Maroon" title="c29">i</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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></span>)</span> &amp; <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="polynom1.html#K5" title="POLYNOM1:func.5">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="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</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> )
 
<b>by </b><i><a class="txt" href="hilbasis.html#E29:37_2_1_1_1"><i><font color="Green" title="E48">A34</font></i></a>, <a class="txt" href="hilbasis.html#E18:37_2_1_1_1_7"><i><font color="Green" title="E60">A85</font></i></a>, <a class="txt" href="hilbasis.html#E20:37_2_1_1_1_7"><i><font color="Green" title="E62">A87</font></i></a></i>;<br/>
<a NAME="E23:37_2_1_1_1_7"/><i><font color="Green" title="E64">A89</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c29">i</font> &amp; <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 )
 
<b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="hilbasis.html#E14:37_2_1_1_1"><i><font color="Green" title="E40">A13</font></i></a>, <a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E24:37_2_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 
<b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>;<br/>
<a NAME="E25:37_2_1_1_1_7"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">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="binarith.html#K3" title="BINARITH:func.3">-'</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>
 
<b>by </b><i><a class="txt" href="hilbasis.html#E23:37_2_1_1_1_7"><i><font color="Green" title="E64">A89</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<b>hence </b><a NAME="E26:37_2_1_1_1_7"/>
<font color="Maroon" title="c27">n1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">p</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:37_2_1_1_1_7"><i><font color="Green" title="E56">A73</font></i></a>, <a class="txt" href="hilbasis.html#E18:37_2_1_1_1_7"><i><font color="Green" title="E60">A85</font></i></a>, <a class="txt" href="hilbasis.html#E19:37_2_1_1_1_7"><i><font color="Green" title="E61">A86</font></i></a>, <a class="txt" href="hilbasis.html#E20:37_2_1_1_1_7"><i><font color="Green" title="E62">A87</font></i></a>, <a class="txt" href="hilbasis.html#E22:37_2_1_1_1_7"><i><font color="Green" title="E63">A88</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E45:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E55">A90</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E46:37_2_1_1_1"/><i><font color="Green" title="E56">A91</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">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="c25">u</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E41:37_2_1_1_1"><i><font color="Green" title="E53">A60</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><i><font color="Green" title="E57">A92</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c25">u</font> = 
 <a href="card_1.html#K1" title="CARD_1:func.1">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="c25">u</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c26">p</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E46:37_2_1_1_1"><i><font color="Green" title="E56">A91</font></i></a>, <a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>
<br/>.= 
 <a href="card_1.html#K1" title="CARD_1:func.1">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E43:37_2_1_1_1"><i><font color="Green" title="E54">A71</font></i></a>, <a class="txt" href="hilbasis.html#E45:37_2_1_1_1"><i><font color="Green" title="E55">A90</font></i></a>, <a class="ref" href="polynom1.html#T7" title="POLYNOM1:th.7">POLYNOM1:7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_circ.html#T21" title="PRE_CIRC:th.21">PRE_CIRC:21</a></i>
;<br/><a NAME="E48:37_2_1_1_1"/><b>then </b><i><font color="Green" title="E58">A93</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">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="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Maroon" title="c23">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E49:37_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c26">p</font> is   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c25">u</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E43:37_2_1_1_1"><i><font color="Green" title="E54">A71</font></i></a>, <a class="txt" href="hilbasis.html#E45:37_2_1_1_1"><i><font color="Green" title="E55">A90</font></i></a>, <a class="ref" href="funct_2.html#T83" title="FUNCT_2:th.83">FUNCT_2:83</a></i>;<br/><b>hence </b><a NAME="E50:37_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">b'</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">b'</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E24:37_2_1_1_1"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="hilbasis.html#E26:37_2_1_1_1"><i><font color="Green" title="E46">A23</font></i></a>, <a class="txt" href="hilbasis.html#E30:37_2_1_1_1"><i><font color="Green" title="E49">A35</font></i></a>, <a class="txt" href="hilbasis.html#E47:37_2_1_1_1"><i><font color="Green" title="E57">A92</font></i></a>, <a class="txt" href="hilbasis.html#E48:37_2_1_1_1"><i><font color="Green" title="E58">A93</font></i></a>, <a class="ref" href="rlvect_2.html#T8" title="RLVECT_2:th.8">RLVECT_2:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><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">x'</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <font color="Maroon" title="c8">y'</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">x'</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</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">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#T18" title="FUNCT_2:th.18">FUNCT_2:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div>
