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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> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</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> ;<br/><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>





<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1"/>
<font color="Maroon" title="c3">a</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>
 ;<br/><a NAME="E2:24_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">n</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>
 <b>by </b><i><a class="ref" href="hilbasis.html#D5" target="_self" title="HILBASIS:def.5">Def5</a></i>;<br/><a NAME="E3:24_1"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="algseq_1.html#T22" title="ALGSEQ_1:th.22">ALGSEQ_1:22</a></i>;<br/><a NAME="E4:24_1"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E2:24_1_1"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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>
 <b>by </b><i><a class="ref" href="hilbasis.html#D5" target="_self" title="HILBASIS:def.5">Def5</a></i>;<br/></div><b>end;</b></div><a NAME="E6:24_1"/><b>then </b>
<font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</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/><a NAME="E7:24_1"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="algseq_1.html#D4" title="ALGSEQ_1:def.4">ALGSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E8:24_1"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></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="txt" href="hilbasis.html#E4:24_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_2"/><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c3">a</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>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:24_2_1"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_1_1"/>
( <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> or <font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_2_1_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:24_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="hilbasis.html#D5" target="_self" title="HILBASIS:def.5">Def5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_2_1_1_2"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">n</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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>
 <b>by </b><i><a class="ref" href="hilbasis.html#D5" target="_self" title="HILBASIS:def.5">Def5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E3:24_2"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</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/><a NAME="E4:24_2"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="algseq_1.html#D4" title="ALGSEQ_1:def.4">ALGSEQ_1:def 4</a></i>;<br/><b>hence </b><a NAME="E5:24_2"/>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
( ( <font color="Maroon" title="c3">a</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> implies  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></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 ) &amp; ( <font color="Maroon" title="c3">a</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> implies  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" title="HILBASIS:func.5">monomial</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&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="txt" href="hilbasis.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a></i>;<br/>


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