<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c3">PRPNR</font> =  <a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">PN1R</font> =  <a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c5">PNR</font> =  <a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c6">CPRPNR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">CPN1R</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">CPNR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">CR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span>] <b>means </b>for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</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">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span>;<br/>
<div><i><font color="Green" title="E33">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c11">p1</font> = <font color="Maroon" title="c10">x</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>] <b>means </b>for <font color="Olive" title="b1">p2</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">R</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/><font color="Olive" title="b1">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> $1</span>)</span>;<br/><div><i><font color="Green" title="E34">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> = <font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default">$1 <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c13">p2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E2:38_1_1_1_1"/><i><font color="Green" title="E35">A3</font></i>: 
for <font color="Olive" title="b1">b</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> holds  <font color="Maroon" title="c13">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c14">p2</font> = <font color="Maroon" title="c13">p2</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> ;<br/><b>reconsider </b><font color="Maroon" title="c15">p2</font> = <font color="Maroon" title="c14">p2</font> as   <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_1_1_1"/>
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or (  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_1"/><i><font color="Green" title="E36">A4</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c16">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">b</font> = <font color="Maroon" title="c16">b</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="txt" href="hilbasis.html#E3:38_1_1_1_1_1_1_1_1"><i><font color="Green" title="E37">A5</font></i></a></i>;<br/><a NAME="E5:38_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c15">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c17">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:38_1_1_1_1_1_1_1_1"><i><font color="Green" title="E37">A5</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E6:38_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:38_1_1_1_1"><i><font color="Green" title="E35">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_1_1_1_1"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:38_1_1_1_1_1_1_1"><i><font color="Green" title="E36">A4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_2"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:38_1_1_1_1_1_1_2"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:38_1_1_1_1_1_1_3"/><i><font color="Green" title="E36">A6</font></i>: 
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c16">Supportp1</font> =  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c17">Supportp2</font> =  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <b>by </b><i><a class="txt" href="hilbasis.html#E1:38_1_1_1_1_1_1_3"><i><font color="Green" title="E36">A6</font></i></a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c17">Supportp2</font>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font>;<br/><div><i><font color="Green" title="E37">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">x</font> be    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c17">Supportp2</font>;<br/><a NAME="E1:38_1_1_1_1_1_1_3_1"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c18">x</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <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="E2:38_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font>
 ;<br/><a NAME="E3:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c15">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:38_1_1_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="hilbasis.html#E1:38_1_1_1_1_1_1_3_1"><i><font color="Green" title="E38">A8</font></i></a></i>;<br/><a NAME="E5:38_1_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c18">x</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_1_1_1_1_3_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c16">Supportp1</font> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c18">x</font>,<font color="Olive" title="b1">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c18">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c17">Supportp2</font>,<font color="Maroon" title="c16">Supportp1</font><b> such that </b><br/><a NAME="E6:38_1_1_1_1_1_1_3"/><i><font color="Green" title="E38">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of <font color="Maroon" title="c17">Supportp2</font> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c18">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E4:38_1_1_1_1_1_1_3"><i><font color="Green" title="E37">A7</font></i></a>);<br/></i><a NAME="E7:38_1_1_1_1_1_1_3"/><i><font color="Green" title="E39">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">Supportp2</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c20">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_1_1_1_3_2"/><i><font color="Green" title="E40">A11</font></i>: 
( <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f</font> &amp; <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x2</font> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">x1'</font> = <font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c22">x2'</font> = <font color="Maroon" title="c20">x2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> <b>by </b><i><a class="txt" href="hilbasis.html#E7:38_1_1_1_1_1_1_3"><i><font color="Green" title="E39">A10</font></i></a></i>;<br/><i><font color="Green" title="E41">A12</font></i>: <font color="Maroon" title="c21">x1'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font> = 
<font color="Maroon" title="c18">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E6:38_1_1_1_1_1_1_3"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="hilbasis.html#E1:38_1_1_1_1_1_1_3_2"><i><font color="Green" title="E40">A11</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c22">x2'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E6:38_1_1_1_1_1_1_3"><i><font color="Green" title="E38">A9</font></i></a>, <a class="txt" href="hilbasis.html#E1:38_1_1_1_1_1_1_3_2"><i><font color="Green" title="E40">A11</font></i></a></i>
;<br/><font color="Maroon" title="c21">x1'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x1'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c22">x2'</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E3:38_1_1_1_1_1_1_3_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
;<br/><b>hence </b><a NAME="E5:38_1_1_1_1_1_1_3_2"/>
<font color="Maroon" title="c19">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font>
 ;<br/></div><b>end;</b></div><a NAME="E9:38_1_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c18">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E10:38_1_1_1_1_1_1_3"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Supportp1</font>
 ;<br/><a NAME="E11:38_1_1_1_1_1_1_3"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c17">Supportp2</font> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c16">Supportp1</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E7:38_1_1_1_1_1_1_3"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="hilbasis.html#E9:38_1_1_1_1_1_1_3"><i><font color="Green" title="E40">A13</font></i></a>, <a class="ref" href="card_1.html#T26" title="CARD_1:th.26">CARD_1:26</a></i>;<br/><b>hence </b><a NAME="E12:38_1_1_1_1_1_1_3"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c15">p2</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_2.html#T68" title="CARD_2:th.68">CARD_2:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E6:38_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c15">p2</font> is <a href="polynom1.html#V2" title="POLYNOM1:attr.2">finite-Support</a>
 <b>by </b><i><a class="ref" href="polynom1.html#D10" title="POLYNOM1:def.10">POLYNOM1:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">p2'</font> = <font color="Maroon" title="c15">p2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>take </b><font color="Maroon" title="c17">p2'</font> = <font color="Maroon" title="c16">p2'</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">p2''</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">R</font>;<br/><b>let </b><font color="Maroon" title="c19">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>;<br/><b>assume </b><a NAME="E1:38_1_1_1_1_2"/><i><font color="Green" title="E36">A14</font></i>: 
<font color="Maroon" title="c18">p2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">p2'</font>
 ;<br/><a NAME="E2:38_1_1_1_1_2"/>
<font color="Maroon" title="c19">b</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <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/><b>hence </b><a NAME="E3:38_1_1_1_1_2"/>
<font color="Maroon" title="c18">p2''</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c12">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E2:38_1_1_1_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="hilbasis.html#E1:38_1_1_1_1_2"><i><font color="Green" title="E36">A14</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:38_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">i</font>,<font color="Maroon" title="c17">p2'</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c12">p3</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E4:38_1_1_1"/><i><font color="Green" title="E35">A15</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c12">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E2:38_1_1_1"><i><font color="Green" title="E34">A2</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c13">p3</font> = <font color="Maroon" title="c12">p3</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1_1_2_1"/>
(  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1_2_1_1"/><i><font color="Green" title="E36">A16</font></i>: 
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c14">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><b>reconsider </b><font color="Maroon" title="c15">i1</font> = <font color="Maroon" title="c14">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">p2</font> = <font color="Maroon" title="c13">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_1_1"/><i><font color="Green" title="E37">A17</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:38_1_1_1_2_1_1_1"/><i><font color="Green" title="E38">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c18">x'</font> = <font color="Maroon" title="c17">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> ;<br/><a NAME="E3:38_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">x'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c14">i</font></span>)</span> <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">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:38_1_1_1_2_1_1"><i><font color="Green" title="E36">A16</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c18">x'</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">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E5:38_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c16">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E7:38_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c16">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1_2_1_1_1"><i><font color="Green" title="E37">A17</font></i></a>, <a class="txt" href="hilbasis.html#E5:38_1_1_1_2_1_1_1"><i><font color="Green" title="E38">A18</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><font color="Maroon" title="c13">p3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> = 
 <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#D24" title="POLYNOM1:def.24">POLYNOM1:def 24</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:38_1_1_1_2_1_1"/>
<font color="Maroon" title="c13">p3</font> is <a href="algseq_1.html#V1" title="ALGSEQ_1:attr.1">finite-Support</a>
 <b>by </b><i><a class="ref" href="algseq_1.html#D2" title="ALGSEQ_1:def.2">ALGSEQ_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_1_2_1_2"/>
 <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c14">Supportp1</font> =  <a href="polynom1.html#K11" title="POLYNOM1:func.11">Support</a> <font color="Maroon" title="c11">p1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <b>-&gt; </b>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  = $1 <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>;<br/><b>consider </b><font color="Maroon" title="c15">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:38_1_1_1_2_1_2"/><i><font color="Green" title="E36">A19</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> holds  <font color="Maroon" title="c15">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i><b>reconsider </b><font color="Maroon" title="c16">j</font> =  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">Supportp1</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_2_1"/><i><font color="Green" title="E37">A20</font></i>: 
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><b>reconsider </b><font color="Maroon" title="c18">i1</font> = <font color="Maroon" title="c17">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">p2</font> = <font color="Maroon" title="c13">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">i1</font> as   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_2_1"/><i><font color="Green" title="E38">A21</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E5:38_1_1_1_2_1_2_1"/><i><font color="Green" title="E39">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1_2_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c21">x'</font> = <font color="Maroon" title="c20">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font> ;<br/><a NAME="E3:38_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">j</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E1:38_1_1_1_2_1_2_1"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:38_1_1_1_2_1_2_1_1"/><b>then </b><i><font color="Green" title="E40">A23</font></i>: 
not <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c14">Supportp1</font>
 <b>by </b><i><a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/><font color="Maroon" title="c15">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c17">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">x'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c17">i</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1_2_1_2"><i><font color="Green" title="E36">A19</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c17">i</font>
<b>by </b><i><a class="ref" href="hilbasis.html#D1" target="_self" title="HILBASIS:def.1">Def1</a></i>
;<br/><a NAME="E6:38_1_1_1_2_1_2_1_1"/><b>then </b>
not <font color="Maroon" title="c21">x'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">Supportp1</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1_2_1_2_1_1"><i><font color="Green" title="E40">A23</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/><a NAME="E7:38_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">x'</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c17">i</font></span>)</span> <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">R</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E8:38_1_1_1_2_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c21">x'</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">R</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/><b>hence </b><a NAME="E9:38_1_1_1_2_1_2_1_1"/>
<font color="Maroon" title="c19">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div><a NAME="E7:38_1_1_1_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c19">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K13" title="POLYNOM1:func.13">Bags</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1_2_1_2_1"><i><font color="Green" title="E38">A21</font></i></a>, <a class="txt" href="hilbasis.html#E5:38_1_1_1_2_1_2_1"><i><font color="Green" title="E39">A22</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><font color="Maroon" title="c13">p3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> = 
 <a href="polynom1.html#K25" title="POLYNOM1:func.25">0_</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="polynom1.html#D24" title="POLYNOM1:def.24">POLYNOM1:def 24</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:38_1_1_1_2_1_2"/>
<font color="Maroon" title="c13">p3</font> is <a href="algseq_1.html#V1" title="ALGSEQ_1:attr.1">finite-Support</a>
 <b>by </b><i><a class="ref" href="algseq_1.html#D2" title="ALGSEQ_1:def.2">ALGSEQ_1:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c13">p3</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12" title="POLYNOM3:def.12">POLYNOM3:def 12</a></i>;<br/><b>take </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c14">y</font>;<br/><b>thus </b><a NAME="E8:38_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c15">y</font>]
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:38_1_1_1"><i><font color="Green" title="E35">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:38_1_1"/><i><font color="Green" title="E34">A24</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="hilbasis.html#E1:38_1_1"><i><font color="Green" title="E33">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">P</font> = <font color="Maroon" title="c10">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom3.html#K13" title="POLYNOM3:func.13">Polynom-Ring</a> <span class="p2">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c11">P</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">p1</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c13">p2</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">R</font>;<br/><b>let </b><font color="Maroon" title="c14">p3</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c16">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>;<br/><b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green" title="E35">A25</font></i>: 
( <font color="Maroon" title="c14">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p1</font> &amp; <font color="Maroon" title="c13">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">i</font> )
 ;<br/><a NAME="E2:38_1_1_2"/>
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>;<br/><b>hence </b><a NAME="E3:38_1_1_2"/>
<font color="Maroon" title="c13">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Maroon" title="c15">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:38_1_1"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="hilbasis.html#E1:38_1_1_2"><i><font color="Green" title="E35">A25</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:38_1_1"/>
for <font color="Olive" title="b1">p1</font> being  <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">p2</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">R</font><br/> for <font color="Olive" title="b3">p3</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K29" title="POLYNOM1:func.29">Polynom-Ring</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">R</font></span>)</span><br/> for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b5">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c2">n</font>  st <font color="Olive" title="b3">p3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p1</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">i</font> holds <br/><font color="Olive" title="b2">p2</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <font color="Olive" title="b5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p1</font> <a href="polynom1.html#K14" title="POLYNOM1:func.14">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">b</font> <a href="hilbasis.html#K1" title="HILBASIS:func.1">bag_extend</a> <font color="Olive" title="b4">i</font></span>)</span>
 ;<br/>


</div>
