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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">Abelian</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>





<b>assume </b><a NAME="E1:49"/><i><font color="Green" title="E36">A1</font></i>: 
 ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">q</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>;<br/>



<b>consider </b><font color="Maroon" title="c6">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:49"/><i><font color="Green" title="E37">A2</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E4:49"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c6">b</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">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="E5:49"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:49"/><b>then </b><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">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="c2">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/>
<a NAME="E7:49"/><i><font color="Green" title="E40">A5</font></i>: 
for <font color="Olive" title="b1">b'</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Olive" title="b1">b'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>

<b>assume </b><a NAME="E1:49_1"/>
<font color="Maroon" title="c7">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E2:49_1"/><b>then </b><i><font color="Green" title="E41">A6</font></i>: 
not <font color="Maroon" title="c7">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:49_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c7">b'</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E2:49_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="polynom2.html#E3:49_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c7">b'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c7">sq</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">spq</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c9">ypq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E9:49"/><i><font color="Green" title="E41">A8</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ypq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c9">ypq</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ypq</font> holds <br/><font color="Maroon" title="c9">ypq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">yq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E11:49"/><i><font color="Green" title="E42">A9</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> &amp;  <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c10">yq</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font> holds <br/><font color="Maroon" title="c10">yq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></i>;<br/>
<a NAME="E12:49"/><i><font color="Green" title="E43">A10</font></i>: 
(  <a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> &amp;  <a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom2.html#T20" target="_self" title="POLYNOM2:th.20">Th20</a></i>;<br/>
<a NAME="E13:49"/><i><font color="Green" title="E44">A11</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#T79" title="POLYNOM1:th.79">POLYNOM1:79</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1"/>
( <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> or  not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1"><b>case </b></a><a NAME="E1:49_2_1_1"/><i><font color="Green" title="E45">A12</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_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="E1:49_2_1_1_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> or <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1"><b>case </b></a><a NAME="E1:49_2_1_1_1_1_1"/><i><font color="Green" title="E46">A13</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
 ;<br/><div class="add"><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
;<br/><a NAME="E3:49_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E47">A14</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:49_2_1_1_1_1_1"/><i><font color="Green" title="E48">A15</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_1_1_1_1_2"/><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_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="E1:49_2_1_1_1_1_1_2_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_1_2"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_1_2_1_1"/><i><font color="Green" title="E50">A17</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1_1_1_2_1_1"/><b>then </b>
not <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49_2_1_1_1_1_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="polynom2.html#E13:49"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_1_2_1_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:49_2_1_1_1_1_1"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_1_1_1_1_3"/><i><font color="Green" title="E49">A18</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1_1_1_1_3_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1_3_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1_1_1_3_1_1"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_1_3_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_1_3_1_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1_1_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/><a NAME="E3:49_2_1_1_1_1_1_3_1_2"/><b>then </b><i><font color="Green" title="E50">A19</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</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="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_1_3"><i><font color="Green" title="E49">A18</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:49_2_1_1_1_1_1_3_1_2"/>
<font color="Maroon" title="c12">u</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">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="E5:49_2_1_1_1_1_1_3_1_2"/><b>then </b>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49_2_1_1_1_1_1_3_1_2"><i><font color="Green" title="E50">A19</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><b>hence </b><a NAME="E6:49_2_1_1_1_1_1_3_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:49_2_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E49">A20</font></i>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>thus </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="polynom2.html#E3:49_2_1_1_1_1_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="polynom2.html#E6:49_2_1_1_1_1_1"><i><font color="Green" title="E49">A20</font></i></a>, <a class="txt" href="polynom2.html#E38"><i><font color="Green" title="E34">Lm10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="polynom2.html#T21" target="_self" title="POLYNOM2:th.21">Th21</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2"><b>case </b></a><a NAME="E1:49_2_1_1_1_1_2"/><i><font color="Green" title="E46">A21</font></i>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2_1_1_1_1_2_1"/><i><font color="Green" title="E47">A22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
 ;<br/>

<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/>
<b>hence </b><a NAME="E3:49_2_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2"><i><font color="Green" title="E46">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:49_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</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="c2">L</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>;<br/><a NAME="E4:49_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E47">A23</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">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="c2">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>;<br/><a NAME="E5:49_2_1_1_1_1_2"/><i><font color="Green" title="E48">A24</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_1_1_1_2_2"/><i><font color="Green" title="E49">A25</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> )
 </span><b>by </b><i><a class="txt" href="polynom2.html#E13:49"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_2"><i><font color="Green" title="E49">A25</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E6:49_2_1_1_1_1_2"/><i><font color="Green" title="E49">A26</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_1_1_1_2_3"/><i><font color="Green" title="E50">A27</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_2_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E4:49"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2"><i><font color="Green" title="E47">A23</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_1_1_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 ;<br/><div class="add"><a NAME="E2:49_2_1_1_1_1_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/><a NAME="E3:49_2_1_1_1_1_2_3_1_2"/><b>then </b><i><font color="Green" title="E51">A28</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</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="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_3"><i><font color="Green" title="E50">A27</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:49_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c12">u</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">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="E5:49_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49_2_1_1_1_1_2_3_1_2"><i><font color="Green" title="E51">A28</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E7:49_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E50">A29</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E5:49_2_1_1_1_1_2"><i><font color="Green" title="E48">A24</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E8:49_2_1_1_1_1_2"/><i><font color="Green" title="E51">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ypq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="polynom2.html#E5:49_2_1_1_1_1_2"><i><font color="Green" title="E48">A24</font></i></a>, <a class="txt" href="polynom2.html#E6:49_2_1_1_1_1_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><a NAME="E9:49_2_1_1_1_1_2"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E12:49"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">TRIANG_1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">k</font> being    <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> <b> such that </b><br/><a NAME="E11:49_2_1_1_1_1_2"/><i><font color="Green" title="E52">A31</font></i>: 
( <font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">sqk</font> being    <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> <b> such that </b><br/><a NAME="E13:49_2_1_1_1_1_2"/><i><font color="Green" title="E53">A32</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c12">sqk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">spqk</font> being    <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> <b> such that </b><br/><a NAME="E15:49_2_1_1_1_1_2"/><i><font color="Green" title="E54">A33</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c13">spqk</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><i><font color="Green" title="E55">A34</font></i>: <font color="Maroon" title="c12">sqk</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E13:49_2_1_1_1_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E5:49_2_1_1_1_1_2"><i><font color="Green" title="E48">A24</font></i></a>, <a class="txt" href="polynom2.html#E6:49_2_1_1_1_1_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>
<br/>.= 
<font color="Maroon" title="c13">spqk</font>
<b>by </b><i><a class="txt" href="polynom2.html#E15:49_2_1_1_1_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/><a NAME="E17:49_2_1_1_1_1_2"/><i><font color="Green" title="E56">A35</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E18:49_2_1_1_1_1_2"/><i><font color="Green" title="E57">A36</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">k</font> &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">sqk</font> )
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="polynom2.html#E13:49_2_1_1_1_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">i</font> being    <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> <b> such that </b><br/><a NAME="E20:49_2_1_1_1_1_2"/><i><font color="Green" title="E58">A37</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/><a NAME="E21:49_2_1_1_1_1_2"/><i><font color="Green" title="E59">A38</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E20:49_2_1_1_1_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E22:49_2_1_1_1_1_2"/><i><font color="Green" title="E60">A39</font></i>: 
<font color="Maroon" title="c12">sqk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="polynom2.html#E13:49_2_1_1_1_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E23:49_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E18:49_2_1_1_1_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="polynom2.html#E20:49_2_1_1_1_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E24:49_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E61">A40</font></i>: 
<font color="Maroon" title="c11">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yq</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E18:49_2_1_1_1_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="polynom2.html#E20:49_2_1_1_1_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E25:49_2_1_1_1_1_2"/><i><font color="Green" title="E62">A41</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">ypq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">sqk</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E15:49_2_1_1_1_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="polynom2.html#E16:49_2_1_1_1_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E26:49_2_1_1_1_1_2"/><i><font color="Green" title="E63">A42</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">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/><a NAME="E27:49_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</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="c4">q</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E28:49_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E64">A43</font></i>: 
<font color="Maroon" title="c11">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E29:49_2_1_1_1_1_2"/><i><font color="Green" title="E65">A44</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">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/><a NAME="E30:49_2_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E31:49_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E66">A45</font></i>: 
<font color="Maroon" title="c11">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><b>then </b><i><font color="Green" title="E67">A46</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="polynom2.html#E31:49_2_1_1_1_1_2"><i><font color="Green" title="E66">A45</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E68">A47</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font>
<b>by </b><i><a class="txt" href="polynom2.html#E28:49_2_1_1_1_1_2"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="polynom2.html#E28:49_2_1_1_1_1_2"><i><font color="Green" title="E64">A43</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
;<br/><i><font color="Green" title="E69">A48</font></i>: <font color="Maroon" title="c9">ypq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E18:49_2_1_1_1_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="polynom2.html#E25:49_2_1_1_1_1_2"><i><font color="Green" title="E62">A41</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:49_2_1_1_1_1_2"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="polynom2.html#E17:49_2_1_1_1_1_2"><i><font color="Green" title="E56">A35</font></i></a>, <a class="txt" href="polynom2.html#E32:49_2_1_1_1_1_2"><i><font color="Green" title="E67">A46</font></i></a>, <a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p0">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E33:49_2_1_1_1_1_2"><i><font color="Green" title="E68">A47</font></i></a>, <a class="ref" href="vectsp_1.html#D18" title="VECTSP_1:def.18">VECTSP_1:def 18</a>, <a class="txt" href="polynom2.html#E17:49_2_1_1_1_1_2"><i><font color="Green" title="E56">A35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">yq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c11">k</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="polynom2.html#E18:49_2_1_1_1_1_2"><i><font color="Green" title="E57">A36</font></i></a>, <a class="txt" href="polynom2.html#E22:49_2_1_1_1_1_2"><i><font color="Green" title="E60">A39</font></i></a></i>
;<br/><a NAME="E35:49_2_1_1_1_1_2"/>
for <font color="Olive" title="b1">i'</font> being   <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>   st <font color="Olive" title="b1">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yq</font> &amp; <font color="Olive" title="b1">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">k</font> holds <br/><font color="Maroon" title="c9">ypq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">yq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">i'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_2_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i'</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_1_1_1_2_8"/><i><font color="Green" title="E70">A49</font></i>: 
( <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">yq</font> &amp; <font color="Maroon" title="c15">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">k</font> )
 ;<br/>

<a NAME="E2:49_2_1_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E71">A50</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i'</font> &amp; <font color="Maroon" title="c15">i'</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">yq</font> )
 
<b>by </b><i><a class="txt" href="polynom2.html#E20:49_2_1_1_1_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="txt" href="polynom2.html#E21:49_2_1_1_1_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:49_2_1_1_1_1_2_8"/>
<font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E8:49_2_1_1_1_1_2"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_8"><i><font color="Green" title="E70">A49</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:49_2_1_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E72">A51</font></i>: 
<font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:49_2_1_1_1_1_2_8"/><i><font color="Green" title="E73">A52</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1_1_1_2_8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_2_1_1_1_1_2_8_1"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 ;<br/>

<a NAME="E2:49_2_1_1_1_1_2_8_1"/><b>then </b><i><font color="Green" title="E74">A53</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E3:49_2_1_1_1_1_2_8_1"/><i><font color="Green" title="E75">A54</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:49_2_1_1_1_1_2"><i><font color="Green" title="E48">A24</font></i></a>, <a class="txt" href="polynom2.html#E6:49_2_1_1_1_1_2"><i><font color="Green" title="E49">A26</font></i></a>, <a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<a NAME="E4:49_2_1_1_1_1_2_8_1"/>
 <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="polynom2.html#E12:49"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="triang_1.html#T8" title="TRIANG_1:th.8">TRIANG_1:8</a></i>;<br/>
<b>hence </b><a NAME="E5:49_2_1_1_1_1_2_8_1"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2"><i><font color="Green" title="E52">A31</font></i></a>, <a class="txt" href="polynom2.html#E13:49_2_1_1_1_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="polynom2.html#E15:49_2_1_1_1_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="polynom2.html#E16:49_2_1_1_1_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_8"><i><font color="Green" title="E70">A49</font></i></a>, <a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="txt" href="polynom2.html#E2:49_2_1_1_1_1_2_8_1"><i><font color="Green" title="E74">A53</font></i></a>, <a class="txt" href="polynom2.html#E3:49_2_1_1_1_1_2_8_1"><i><font color="Green" title="E75">A54</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:49_2_1_1_1_1_2_8"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:49_2_1_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E74">A55</font></i>: 
<font color="Maroon" title="c15">i'</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E29:49_2_1_1_1_1_2"><i><font color="Green" title="E65">A44</font></i></a>, <a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E75">A56</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:49_2_1_1_1_1_2_8"><i><font color="Green" title="E74">A55</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<a NAME="E9:49_2_1_1_1_1_2_8"/><i><font color="Green" title="E76">A57</font></i>: 
<font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="polynom2.html#E20:49_2_1_1_1_1_2"><i><font color="Green" title="E58">A37</font></i></a>, <a class="txt" href="polynom2.html#E21:49_2_1_1_1_1_2"><i><font color="Green" title="E59">A38</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_1_1_1_2_8"><i><font color="Green" title="E70">A49</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:49_2_1_1_1_1_2_8"/>
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E13:49_2_1_1_1_1_2"><i><font color="Green" title="E53">A32</font></i></a>, <a class="txt" href="polynom2.html#E15:49_2_1_1_1_1_2"><i><font color="Green" title="E54">A33</font></i></a>, <a class="txt" href="polynom2.html#E16:49_2_1_1_1_1_2"><i><font color="Green" title="E55">A34</font></i></a>, <a class="txt" href="polynom2.html#E4:49_2_1_1_1_1_2_8"><i><font color="Green" title="E72">A51</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E11:49_2_1_1_1_1_2_8"/><b>then </b><i><font color="Green" title="E77">A58</font></i>: 
<font color="Maroon" title="c15">i'</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E26:49_2_1_1_1_1_2"><i><font color="Green" title="E63">A42</font></i></a>, <a class="txt" href="polynom2.html#E9:49_2_1_1_1_1_2_8"><i><font color="Green" title="E76">A57</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/>
<b>then </b><i><font color="Green" title="E78">A59</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E11:49_2_1_1_1_1_2_8"><i><font color="Green" title="E77">A58</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E9:49_2_1_1_1_1_2_8"><i><font color="Green" title="E76">A57</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c9">ypq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E8:49_2_1_1_1_1_2"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="polynom2.html#E2:49_2_1_1_1_1_2_8"><i><font color="Green" title="E71">A50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="polynom2.html#E7:49_2_1_1_1_1_2"><i><font color="Green" title="E50">A29</font></i></a>, <a class="txt" href="polynom2.html#E5:49_2_1_1_1_1_2_8"><i><font color="Green" title="E73">A52</font></i></a>, <a class="txt" href="polynom2.html#E8:49_2_1_1_1_1_2_8"><i><font color="Green" title="E75">A56</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">yq</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c15">i'</font>
<b>by </b><i><a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="polynom2.html#E2:49_2_1_1_1_1_2_8"><i><font color="Green" title="E71">A50</font></i></a>, <a class="txt" href="polynom2.html#E12:49_2_1_1_1_1_2_8"><i><font color="Green" title="E78">A59</font></i></a></i>
;<br/>


</div><b>end;</b></div><b>hence </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c10">yq</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E9:49"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="polynom2.html#E8:49_2_1_1_1_1_2"><i><font color="Green" title="E51">A30</font></i></a>, <a class="txt" href="polynom2.html#E24:49_2_1_1_1_1_2"><i><font color="Green" title="E61">A40</font></i></a>, <a class="txt" href="polynom2.html#E34:49_2_1_1_1_1_2"><i><font color="Green" title="E69">A48</font></i></a>, <a class="ref" href="polynom2.html#T7" target="_self" title="POLYNOM2:th.7">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="polynom2.html#E11:49"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="polynom2.html#T21" target="_self" title="POLYNOM2:th.21">Th21</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:49_2_1_1"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2"><b>case </b></a><a NAME="E1:49_2_1_2"/><i><font color="Green" title="E45">A60</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/><div class="add"><i><font color="Green" title="E46">A61</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#D21" title="POLYNOM1:def.21">POLYNOM1:def 21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E4:49"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_2"><i><font color="Green" title="E45">A60</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/><a NAME="E3:49_2_1_2"/><i><font color="Green" title="E47">A62</font></i>: 
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E13:49"><i><font color="Green" title="E44">A11</font></i></a></i>;<br/><a NAME="E4:49_2_1_2"/>
for <font color="Olive" title="b1">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span> holds <br/><font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:49_2_1_2_2"/><i><font color="Green" title="E48">A63</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_2_1_2_2_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font> or <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font> )
 </span><b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_2_2"><i><font color="Green" title="E48">A63</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:49_2_1_2_2_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font>
 ;<br/><div class="add"><a NAME="E2:49_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_2_2_1_1"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E4:49"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="polynom2.html#E6:49"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="polynom2.html#E2:49_2_1_2"><i><font color="Green" title="E46">A61</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:49_2_1_2_2_1_2"/><i><font color="Green" title="E49">A64</font></i>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E3:49_2_1_2_2_1_2"/><i><font color="Green" title="E50">A65</font></i>: 
<font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</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="c2">L</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_2_2_1_2"><i><font color="Green" title="E49">A64</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:49_2_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q</font> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c12">u</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_2"><i><font color="Green" title="E45">A60</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_2_2_1_2"><i><font color="Green" title="E49">A64</font></i></a></i>;<br/><b>hence </b><a NAME="E5:49_2_1_2_2_1_2"/>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:49_2_1_2_2_1_2"><i><font color="Green" title="E49">A64</font></i></a>, <a class="txt" href="polynom2.html#E3:49_2_1_2_2_1_2"><i><font color="Green" title="E50">A65</font></i></a>, <a class="ref" href="polynom1.html#D9" title="POLYNOM1:def.9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:49_2_1_2"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="polynom2.html#E3:49_2_1_2"><i><font color="Green" title="E47">A62</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/><b>hence </b> <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span> <a href="polynom1.html#K15" title="POLYNOM1:func.15">.</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E7:49"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="polynom2.html#E1:49_2_1_2"><i><font color="Green" title="E45">A60</font></i></a>, <a class="txt" href="polynom2.html#E38"><i><font color="Green" title="E34">Lm10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E3:49"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="polynom2.html#E2:49_2_1_2"><i><font color="Green" title="E46">A61</font></i></a>, <a class="ref" href="polynom2.html#T21" target="_self" title="POLYNOM2:th.21">Th21</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:49"/>
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K23" title="POLYNOM1:func.23">+</a> <font color="Maroon" title="c4">q</font></span>)</span>,<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">x</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">x</font></span>)</span>
 ;<br/>


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