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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  <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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V6">right_unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_1.html#V9">Field-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <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">R</font> be    <a href="rewrite1.html#M1">RedSequence</a> of  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>











<b>assume </b><a NAME="E1:55"/><i><font color="Green">E39</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 ;<br/>

<a NAME="E2:55"/><b>then </b><i><font color="Green">E40</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">R</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:55_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_1"><b>case </b></a><a NAME="E1:55_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T7" target="_self">Th7</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:55_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:55_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E5:55_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E6:55_1_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/><a NAME="E7:55_1_1_1"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E8:55_1_1_1"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="55_1_1_2"><b>case </b></a><a NAME="E1:55_1_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 ;<br/><div class="add"><a NAME="E2:55_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="groeb_3.html#T7" target="_self">Th7</a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/><a NAME="E3:55_1_1_2"/>
1 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="groeb_3.html#T7" target="_self">Th7</a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E4:55_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E5:55_1_1_2"/>
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T148">XREAL_1:148</a></i>;<br/><a NAME="E6:55_1_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">R</font>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:55_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E8:55_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font>
 <b>by </b><i>, , <a class="ref" href="rewrite1.html#D2">REWRITE1:def 2</a></i>;<br/><a NAME="E9:55_1_1_2"/><b>then </b>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="polyred.html#K3">PolyRedRel</a> <font color="Maroon">P</font>,<font color="Maroon">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E10:55_1_1_2"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:55"/>
<font color="Maroon">R</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>
 ;<br/>


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