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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">Z</font> =  <a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">b</font> be    <a href="polynom1.html#M1" target="_self">Element</a> of  <a href="polynom1.html#K14" target="_self">Bags</a> <font color="Maroon">n</font>;<br/><b>consider </b><font color="Maroon">s</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E2:176_1"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E3:176_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span>
 <b>and </b><br/><a NAME="E4:176_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font> holds <br/> ex <font color="Olive">b1</font>, <font color="Olive">b2</font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font> st <br/>( <span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Olive">b1</font>,<font color="Olive">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span> &amp; <font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Olive">b2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="polynom1.html#T24" target="_self">Th24</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:176_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">s</font>
 ;<br/><b>then consider </b><font color="Maroon">b1</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>, <font color="Maroon">b2</font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:176_1_1"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K21" target="_self">decomp</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K3" target="_self">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K10">&lt;*</a><span class="default"><font color="Maroon">b1</font>,<font color="Maroon">b2</font></span><a href="finseq_1.html#K10">*&gt;</a></span>
 <b>and </b><br/><a NAME="E4:176_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b2</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#T29" target="_self">Th29</a></i>;<br/><b>thus </b><font color="Maroon">s</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b1</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T30" target="_self">Th30</a>, <a class="ref" href="polynom1.html#T21" target="_self">Th21</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36">VECTSP_1:36</a></i>
;<br/></div><b>end;</b></div><a NAME="E6:176_1"/><b>then </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="matrlin.html#T15">MATRLIN:15</a></i>;<br/><b>hence </b><a NAME="E7:176_1"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <span class="p2">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="polynom1.html#K15" target="_self">.</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:176"/>
<font color="Maroon">p</font> <a href="polynom1.html#K28" target="_self">*'</a> <span class="p1">(<span class="default"><a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="polynom1.html#K26" target="_self">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


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