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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</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/>



<a NAME="E1:20"/><i><font color="Green">E36</font></i>: 
 <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:20_1_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/><a NAME="E4:20_1_1"/><i><font color="Green">E53</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="algseq_1.html#D3">ALGSEQ_1:def 3</a><br/>

<b>assume </b><a NAME="E1:20_1_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:20_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" 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="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E3:20_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E4:20_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E5:20_1_1"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E6:20_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:20_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">m</font> <a href="algseq_1.html#R1">is_at_least_length_of</a>  <a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:20_1_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>assume </b><a NAME="E3:20_1_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E4:20_1_1_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:20_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K6">-</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="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E6:20_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E7:20_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</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#D3">ALGSEQ_1:def 3</a></i>;<br/><a NAME="E8:20_1_1_2"/><b>then </b>
<font color="Maroon">p</font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>hence </b><a NAME="E9:20_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="algseq_1.html#T25">ALGSEQ_1:25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:20_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="algseq_1.html#D4">ALGSEQ_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/><i><font color="Green">E61</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b> <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom4.html#K1" target="_self">Leading-Monomial</a> <font color="Maroon">p</font></span>)</span> = 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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