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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a>  <a href="vectsp_1.html#L2" title="VECTSP_1:struct.2">multLoopStr_0</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <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="c3">p</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">ps</font> =  <a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">lpn</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">n</font>;<br/>
<a NAME="E2:61"/>
<font color="Maroon" title="c2">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="uproots.html#E1:61"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">lpn</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c2">n</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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c6">lpn</font>
 ;<br/><a NAME="E2:61_1"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c2">n</font> <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="c3">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c3">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="ref" href="uproots.html#D6" target="_self" title="UPROOTS:def.6">Def6</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="uproots.html#E2:61_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:61"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c6">lpn</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D3" title="ALGSEQ_1:def.3">ALGSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_2"/><i><font color="Green" title="E50">A4</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:61_2"/><i><font color="Green" title="E51">A5</font></i>: 
<font color="Maroon" title="c6">lpn</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">m</font>
 ;<br/><a NAME="E3:61_2"/>
<font color="Maroon" title="c6">lpn</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E2:61_2"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:61_2"/><b>then </b><i><font color="Green" title="E52">A6</font></i>: 
<font color="Maroon" title="c6">lpn</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</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>then reconsider </b><font color="Maroon" title="c8">lpn1</font> = <font color="Maroon" title="c6">lpn</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 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>  <b>by </b><i><a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E6:61_2"/><i><font color="Green" title="E53">A7</font></i>: 
<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> <font color="Maroon" title="c8">lpn1</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>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font>
 ;<br/><a NAME="E7:61_2"/><i><font color="Green" title="E54">A8</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c8">lpn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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> <font color="Maroon" title="c8">lpn1</font></span>)</span>
 <b>by </b><i><a class="ref" href="uproots.html#D6" target="_self" title="UPROOTS:def.6">Def6</a></i>;<br/><a NAME="E8:61_2"/>
<font color="Maroon" title="c3">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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> <font color="Maroon" title="c8">lpn1</font></span>)</span> <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>
 <b>by </b><i><a class="txt" href="uproots.html#E6:61_2"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="algseq_1.html#T25" title="ALGSEQ_1:th.25">ALGSEQ_1:25</a></i>;<br/><b>hence </b><a NAME="E9:61_2"/>
contradiction
 <b>by </b><i><a class="txt" href="uproots.html#E1:61_2"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="uproots.html#E4:61_2"><i><font color="Green" title="E52">A6</font></i></a>, <a class="txt" href="uproots.html#E7:61_2"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="algseq_1.html#D3" title="ALGSEQ_1:def.3">ALGSEQ_1:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><span class="p1">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K5" title="UPROOTS:func.5">poly_shift</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font> = 
<font color="Maroon" title="c6">lpn</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="uproots.html#E5:61"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="algseq_1.html#D4" title="ALGSEQ_1:def.4">ALGSEQ_1:def 4</a></i>
<br/>.= 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">p</font>

;<br/><br/>


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