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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="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b2">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> holds <br/>( <font color="Olive" title="b1">p</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> &amp; <font color="Olive" title="b2">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> )</span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> implies ( <font color="Maroon" title="c2">p</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> &amp; <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> ) )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:50"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>
 <b>and </b><br/><a NAME="E2:50"/><i><font color="Green" title="E37">A2</font></i>: 
(  not <font color="Maroon" title="c2">p</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> or  not <font color="Maroon" title="c3">q</font> is <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:50"/>
(  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E2:50"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>;<br/>
<a NAME="E4:50"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T35" target="_self" title="UPROOTS:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:50"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="uproots.html#T19" target="_self" title="UPROOTS:th.19">Th19</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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