<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="vectsp_1.html#V10">degenerated</a>  <a href="vectsp_1.html#L2">multLoopStr_0</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:61"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>

<b>set </b><font color="Maroon">ps</font> =  <a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">lpn</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font>;<br/>
<a NAME="E2:61"/>
<font color="Maroon">n</font> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">lpn</font> = <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">lpn</font>
 ;<br/><a NAME="E2:61_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">i</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="uproots.html#E5:61"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i>, <a class="ref" href="algseq_1.html#T22">ALGSEQ_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E5:61"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">lpn</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="algseq_1.html#D3">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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:61_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:61_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">lpn</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E3:61_2"/>
<font color="Maroon">lpn</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="txt" href="uproots.html#E7:61_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:61_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">lpn</font> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><b>reconsider </b><font color="Maroon">lpn1</font> = <font color="Maroon">lpn</font> <a href="binop_2.html#K10">-</a> 1 as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/><a NAME="E6:61_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lpn1</font></span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E7:61_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">lpn1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lpn1</font></span>)</span>
 <b>by </b><i><a class="txt" href="uproots.html#E5:61"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:61_2"/>
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">lpn1</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T25">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#E6:61_2"><i><font color="Green">E50</font></i></a>, , , <a class="ref" href="algseq_1.html#D3">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#K3">len</a> <span class="p2">(<span class="default"><a href="uproots.html#K5" target="_self">poly_shift</a> <font color="Maroon">p</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font> = 
<font color="Maroon">lpn</font> <a href="binop_2.html#K23">+</a> <font color="Maroon">n</font>
<b>by </b><i>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>

;<br/><br/>


</div>
