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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> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</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">n</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">p</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">L</font>  st  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">L</font></span><br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</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> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font>  st  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span><br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<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"> (  <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font> )</span><br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E2:38"/><i><font color="Green" title="E29">A2</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">p</font> &amp;  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:38"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font> holds <br/><font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</font> implies <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> )</span><br/>

<b>assume </b><a NAME="E1:38_1"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Bags</a> <font color="Maroon" title="c2">n</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">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:38_1"/>
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <font color="Maroon" title="c5">b</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="#E1:38"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="#E1:38_1"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1"/>
<font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">u</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81" title="POLYNOM1:th.81">POLYNOM1:81</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:38"/>
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom1.html#K24" title="POLYNOM1:func.24">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="#E2:38"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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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