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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="vectsp_1.html#V10" title="VECTSP_1:attr.10">degenerated</a>  <a href="vectsp_1.html#L2" title="VECTSP_1:struct.2">multLoopStr_0</a> ;<br/>

<a NAME="E1:7"/>
for <font color="Olive" title="b1">i</font> being   <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>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/><span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="polynom3.html#T31" title="POLYNOM3:th.31">POLYNOM3:31</a></i>;<br/>
<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E7">A1</font></i>: 
1 <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</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="7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i</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>assume </b><b>that </b><br/><a NAME="E1:7_1"/><i><font color="Green" title="E8">A2</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="algseq_1.html#R1" title="ALGSEQ_1:pred.1">is_at_least_length_of</a>  <a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font>
 <b>and </b><br/><a NAME="E2:7_1"/><i><font color="Green" title="E9">A3</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><a NAME="E3:7_1"/><i><font color="Green" title="E10">A4</font></i>: 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#D21" title="VECTSP_1:def.21">VECTSP_1:def 21</a></i>;<br/><a NAME="E4:7_1"/>
0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E2:7_1"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E5:7_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom4.html#E1:7_1"><i><font color="Green" title="E8">A2</font></i></a>, <a class="ref" href="algseq_1.html#D3" title="ALGSEQ_1:def.3">ALGSEQ_1:def 3</a></i>;<br/><b>hence </b><a NAME="E6:7_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom4.html#E3:7_1"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="polynom3.html#T31" title="POLYNOM3:th.31">POLYNOM3:31</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:7"/>
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="polynom4.html#E2:7"><i><font color="Green" title="E7">A1</font></i></a>, <a class="ref" href="algseq_1.html#D4" title="ALGSEQ_1:def.4">ALGSEQ_1:def 4</a></i>;<br/>


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