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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="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="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</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">I</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span><br/> for <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</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="b2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbasis.html#K4" title="HILBASIS:func.4">minlen</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b3">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> holds <br/>( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b3">i2</font> )</span><br/><b>let </b><font color="Maroon" title="c2">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <font color="Maroon" title="c1">L</font></span>)</span>; <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">i1</font>, <font color="Olive" title="b2">i2</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">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbasis.html#K4" title="HILBASIS:func.4">minlen</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/>( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">i2</font> )</span><br/><b>let </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">i2</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="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbasis.html#K4" title="HILBASIS:func.4">minlen</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> implies ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">i2</font> ) )</span><br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbasis.html#K4" title="HILBASIS:func.4">minlen</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c4">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">i2</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c5">i1'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:22"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c5">i1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font> &amp; ( for <font color="Olive" title="b1">x'</font>, <font color="Olive" title="b2">y'</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">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i1'</font> &amp; <font color="Olive" title="b2">y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> holds <br/> <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">x'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b2">y'</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:22"/>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp;  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c4">i2</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:22"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="hilbasis.html#E3:22"><i><font color="Green" title="E20">A2</font></i></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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