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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b2">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> or <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> or <font color="Olive" title="b2">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E2:37"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E3:37"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">lp1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:37"/><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">lp1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a>, <a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></span>;<br/>
<a NAME="E6:37"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></span>;<br/>
<a NAME="E7:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A5</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">lq1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:37"/><span class="lab"><font color="Green" title="E29">A6</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">lq1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:37"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a>, <a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">lp1</font> = <font color="Maroon" title="c4">lp1</font>, <font color="Maroon" title="c7">lq1</font> = <font color="Maroon" title="c5">lq1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E11:37"/><span class="lab"><font color="Green" title="E30">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">lp1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="algseq_1.html#T25" title="ALGSEQ_1:th.25">ALGSEQ_1:25</a></span>;<br/>
<a NAME="E12:37"/><span class="lab"><font color="Green" title="E31">A8</font></span>: 
<font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">lq1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="ref" href="algseq_1.html#T25" title="ALGSEQ_1:th.25">ALGSEQ_1:25</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">lpq</font> = <font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">r</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E14:37"/><span class="lab"><font color="Green" title="E32">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E15:37"/><span class="lab"><font color="Green" title="E33">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</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="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E16:37"/><span class="lab"><font color="Green" title="E34">A11</font></span>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">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="c9">r</font> holds <br/><font color="Maroon" title="c9">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="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom3.html#D11" title="POLYNOM3:def.11">POLYNOM3:def 11</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A12</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">lq1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c7">lq1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/>
<a NAME="E18:37"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E19:37"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A13</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</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="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:37"><span class="lab"><font color="Green" title="E32">A9</font></span></a>, <a class="txt" href="#E7:37"><span class="lab"><font color="Green" title="E28">A5</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">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="c9">r</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> implies <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:37_2"/><span class="lab"><font color="Green" title="E37">A14</font></span>: 
<font color="Maroon" title="c10">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="c9">r</font>
 <span class="kw">and </span><br/><a NAME="E2:37_2"/><span class="lab"><font color="Green" title="E38">A15</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c11">k1</font> = <font color="Maroon" title="c10">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E4:37_2"/><span class="lab"><font color="Green" title="E39">A16</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">k1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c11">k1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E1:37_2"><span class="lab"><font color="Green" title="E37">A14</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_2_1"/>
( <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_2"><span class="lab"><font color="Green" title="E38">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">d</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:37_2_1_1"/><span class="lab"><font color="Green" title="E40">A17</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">and </span><br/><a NAME="E4:37_2_1_1"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#T1" title="FSM_1:th.1">FSM_1:1</a></span>;<br/><a NAME="E5:37_2_1_1"/><span class="lab"><font color="Green" title="E42">A19</font></span>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">lq1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:37"><span class="lab"><font color="Green" title="E29">A6</font></span></a>, <a class="txt" href="#E4:37_2_1_1"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">lq1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">d</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E3:37_2_1_1"><span class="lab"><font color="Green" title="E40">A17</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">lq1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/><span class="kw">hence </span><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="txt" href="#E5:37_2_1_1"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_2_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span><br/><div class="add"><a NAME="E2:37_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:37_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a></span>;<br/><a NAME="E4:37_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c10">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:37_2"><span class="lab"><font color="Green" title="E39">A16</font></span></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span> <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c9">r</font> = 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:37"><span class="lab"><font color="Green" title="E36">A13</font></span></a>, <a class="ref" href="matrix_3.html#T14" title="MATRIX_3:th.14">MATRIX_3:14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">lp1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">lq1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:37"><span class="lab"><font color="Green" title="E34">A11</font></span></a>, <a class="txt" href="#E19:37"><span class="lab"><font color="Green" title="E36">A13</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c6">lp1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c7">lq1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:37"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="txt" href="#E17:37"><span class="lab"><font color="Green" title="E35">A12</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>
;<br/>
<a NAME="E22:37"/><span class="kw">then </span>
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:37"><span class="lab"><font color="Green" title="E30">A7</font></span></a>, <a class="txt" href="#E12:37"><span class="lab"><font color="Green" title="E31">A8</font></span></a>, <a class="ref" href="vectsp_2.html#D5" title="VECTSP_2:def.5">VECTSP_2:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:37"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E15:37"><span class="lab"><font color="Green" title="E33">A10</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
