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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">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<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> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">g</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>, <font color="Maroon">B</font> be    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>













<b>assume </b><a NAME="E1:56"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">A</font> <a href="groeb_2.html#R3" target="_self">is_Standard_Representation_of</a> <font color="Maroon">f</font>,<font color="Maroon">P</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font> &amp; <font color="Maroon">B</font> <a href="groeb_2.html#R3" target="_self">is_Standard_Representation_of</a> <font color="Maroon">g</font>,<font color="Maroon">P</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font> )
 ;<br/>

<a NAME="E2:56"/><b>then </b>
( <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <font color="Maroon">f</font> <a href="polynom1.html#K22">+</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">A</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:56_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_2_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> or  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> ) )
 </span><b>by </b><i><a class="txt" href="groeb_2.html#E1:56"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_1"><b>case </b></a><a NAME="E1:56_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">m</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:56_2_1_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:56"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E5:56_2_1_1_1"/>
 ex <font color="Olive">m</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> ex <font color="Olive">p</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:56"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_2_1_1_2"><b>case </b></a><a NAME="E1:56_2_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:56_2_1_1_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> )
 ;<br/><b>consider </b><font color="Maroon">m</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p</font> being  <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:56_2_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 <b>by </b><i>, , </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="groeb_2.html#E1:56"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E7:56_2_1_1_2"/>
 ex <font color="Olive">m</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> ex <font color="Olive">p</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 <b>by </b><i><a class="txt" href="groeb_2.html#E3:56"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:56_2"/>
 ex <font color="Olive">m</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> ex <font color="Olive">p</font> being <a href="polynom7.html#V1">non-zero</a> <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp;  <a href="termord.html#K3">HT</a> <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span>,<font color="Maroon">T</font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">b</font>,<font color="Maroon">T</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:56"/>
<font color="Maroon">A</font> <a href="groeb_2.html#K5" target="_self">^</a> <font color="Maroon">B</font> <a href="groeb_2.html#R3" target="_self">is_Standard_Representation_of</a> <font color="Maroon">f</font> <a href="polynom1.html#K22">+</a> <font color="Maroon">g</font>,<font color="Maroon">P</font>,<font color="Maroon">b</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/>


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