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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">L</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> <br/>  for <font color="Olive" title="b2">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font>  st  ex <font color="Olive" title="b3">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Olive" title="b2">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="b3">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> for <font color="Olive" title="b3">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font><br/>  for <font color="Olive" title="b4">x</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">L</font> holds   <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b3">q</font></span>)</span>,<font color="Olive" title="b4">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="Olive" title="b2">p</font>,<font color="Olive" title="b4">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="Olive" title="b3">q</font>,<font color="Olive" title="b4">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">L</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>   <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a>   <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a>   <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>   <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>  st  ex <font color="Olive" title="b2">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Olive" title="b1">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="b2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> for <font color="Olive" title="b2">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b3">x</font> being   <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> holds   <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b2">q</font></span>)</span>,<font color="Olive" title="b3">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="Olive" title="b1">p</font>,<font color="Olive" title="b3">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="Olive" title="b2">q</font>,<font color="Olive" title="b3">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K2" title="POLYNOM1:func.2">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> implies  for <font color="Olive" title="b1">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">x</font> being   <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> holds   <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#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b1">q</font>,<font color="Olive" title="b2">x</font>)</span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:43"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> st  <a href="polynom1.html#K2" title="POLYNOM1:func.2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">q</font> being   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font><br/>  for <font color="Olive" title="b2">x</font> being   <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> holds   <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#K5" title="POLYNOM1:func.5">+</a> <font color="Olive" title="b1">q</font></span>)</span>,<font color="Olive" title="b2">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="Olive" title="b2">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="Olive" title="b1">q</font>,<font color="Olive" title="b2">x</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>
<a NAME="E4:43"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">q</font> be   <a href="polynom1.html#NM2" title="POLYNOM1:NM.2">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <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> holds   <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Olive" title="b1">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="Olive" title="b1">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="c5">q</font>,<font color="Olive" title="b1">x</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span><br/>



<a NAME="E6:43"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 for <font color="Olive" title="b1">b9</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>  st <font color="Olive" title="b1">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Olive" title="b1">b9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">b9</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font> )</span><br/>

<a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c7">b9</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:43_1"/>
<font color="Maroon" title="c7">b9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font></span><br/>

<a NAME="E3:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A5</font></span>: 
 not <font color="Maroon" title="c7">b9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c3">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c7">b9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">set </span><font color="Maroon" title="c7">sq</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>);<br/>
<span class="kw">set </span><font color="Maroon" title="c8">spq</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>);<br/>
<a NAME="E7:43"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c4">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/>
<a NAME="E8:43"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T20" title="POLYNOM1:th.20">POLYNOM1:20</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">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><span class="kw"> such that </span><br/><a NAME="E10:43"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:43"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">yq</font>
 <span class="kw">and </span><br/><a NAME="E12:43"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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">yq</font> holds <br/><font color="Maroon" title="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p4">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c6">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">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><span class="kw"> such that </span><br/><a NAME="E14:43"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:43"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
 <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c10">ypq</font>
 <span class="kw">and </span><br/><a NAME="E16:43"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   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">ypq</font> holds <br/><font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p4">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</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="c6">x</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#D4" target="_self" title="POLYNOM2:def.4">Def4</a></span>;<br/>
<a NAME="E17:43"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
 <a href="pre_poly.html#K17" title="PRE_POLY: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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T18" target="_self" title="POLYNOM2:th.18">Th20</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1"/>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> or  not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1"><span class="kw">case </span></a><a NAME="E1:43_2_1_1"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1_1_1_1"/>
( <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> or <font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1"><span class="kw">case </span></a><a NAME="E1:43_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c5">q</font>,<font color="Maroon" title="c6">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="c6">x</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
 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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> implies <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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_2_1_1_1_1_1_1_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_1_1_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="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/><a NAME="E3:43_2_1_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E4:43_2_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c12">u</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E5:43_2_1_1_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E6:43_2_1_1_1_1_1_1_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
;<br/><a NAME="E4:43_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E5:43_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> <span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<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#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1_1_1_1_1_3_1"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_1_3_1_1"/><span class="kw">then </span>
 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="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_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>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_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>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:43_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">thus </span> <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
<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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p4">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E4:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="txt" href="#E6:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="polynom2.html#E36"><span class="lab"><font color="Green" title="E34">Lm10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">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#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="vectsp_1.html#T9" title="VECTSP_1:th.9">VECTSP_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">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="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="polynom2.html#T19" target="_self" title="POLYNOM2:th.19">Th21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2"><span class="kw">case </span></a><a NAME="E1:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A24</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E50">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A25</font></span></a>, <a class="ref" href="rlvect_1.html#D3" title="RLVECT_1:def.3">RLVECT_1:def 3</a></span>
<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="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:43_2_1_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E49">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:43_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T5" title="RLVECT_1:th.5">RLVECT_1:5</a></span>;<br/><a NAME="E4:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>;<br/><a NAME="E5:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
 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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> implies <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_2_2"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">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="c4">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E4:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_2_2_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="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/><div class="add"><a NAME="E2:43_2_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a></span>;<br/><a NAME="E3:43_2_1_1_1_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_1_1_1_2_2"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E4:43_2_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c12">u</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2_1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E6:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
 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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> implies <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_2_3"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_2_1_1_1_1_2_3_1"/>
( <font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_2_3"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_2_3_1_1"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E1:43_2_1_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1_1_2_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_1_1_1_2_3_1_2"/>
<font color="Maroon" title="c12">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E7:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A32</font></span>: 
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><a NAME="E8:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A33</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">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="c9">yq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E6:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">spqk</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A34</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c11">spqk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><a NAME="E11:43_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E1:43_2_1_1"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E13:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A35</font></span>: 
<font color="Maroon" title="c12">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">sqk</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E16:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A37</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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="c13">sqk</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">k</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">k</font> = <font color="Maroon" title="c14">k</font>, <font color="Maroon" title="c16">sqk</font> = <font color="Maroon" title="c13">sqk</font>, <font color="Maroon" title="c17">spqk</font> = <font color="Maroon" title="c11">spqk</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E19:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A38</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E20:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A39</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E21:43_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E22:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A40</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A41</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="txt" href="#E22:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A40</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/><a NAME="E24:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E63">A42</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">sqk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E25:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A43</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E26:43_2_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><a NAME="E27:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A44</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A45</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="txt" href="#E27:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E65">A44</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c18">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E30:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A46</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">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="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E68">A47</font></span>: <font color="Maroon" title="c16">sqk</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>
<br/>.= 
<font color="Maroon" title="c17">spqk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/><a NAME="E32:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A48</font></span>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><a NAME="E33:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A49</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">yq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E34:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A50</font></span>: 
 for <font color="Olive" title="b1">i9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i9</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="c9">yq</font> &amp; <font color="Olive" title="b1">i9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">k</font> holds <br/><font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">i9</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">i9</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="c9">yq</font> &amp; <font color="Maroon" title="c19">i9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">k</font> implies <font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E72">A51</font></span>: 
<font color="Maroon" title="c19">i9</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="c9">yq</font>
 <span class="kw">and </span><br/><a NAME="E2:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E73">A52</font></span>: 
<font color="Maroon" title="c19">i9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span><br/>

<a NAME="E3:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E74">A53</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:43_2_1_1_1_1_2_7"/>
<font color="Maroon" title="c19">i9</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E8:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:43_2_1_1_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A54</font></span>: 
<font color="Maroon" title="c19">i9</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:43_2_1_1_1_1_2_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E7:43_2_1_1_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A55</font></span>: 
<font color="Maroon" title="c19">i9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E77">A56</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c19">i9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E76">A55</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<a NAME="E9:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E78">A57</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_1_1_1_2_7_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_2_1_1_1_1_2_7_2"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:43_2_1_1_1_1_2_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A58</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E3:43_2_1_1_1_1_2_7_2"/><span class="lab"><font color="Green" title="E80">A59</font></span>: 
 <a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>) is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom2.html#T18" target="_self" title="POLYNOM2:th.18">Th20</a>, <a class="ref" href="pre_poly.html#T10" title="PRE_POLY:th.10">PRE_POLY:10</a></span>;<br/>
<a NAME="E4:43_2_1_1_1_1_2_7_2"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:43_2_1_1_1_1_2_7_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E10:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E31:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E2:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E73">A52</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="txt" href="#E2:43_2_1_1_1_1_2_7_2"><span class="lab"><font color="Green" title="E79">A58</font></span></a>, <a class="txt" href="#E3:43_2_1_1_1_1_2_7_2"><span class="lab"><font color="Green" title="E80">A59</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E79">A60</font></span>: 
<font color="Maroon" title="c19">i9</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="txt" href="#E33:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:43_2_1_1_1_1_2_7"/>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="txt" href="#E10:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E31:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="txt" href="#E5:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E75">A54</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<a NAME="E12:43_2_1_1_1_1_2_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A61</font></span>: 
<font color="Maroon" title="c19">i9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E64">A43</font></span></a>, <a class="txt" href="#E10:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E79">A60</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E81">A62</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c5">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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E80">A61</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p3">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E79">A60</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<a NAME="E14:43_2_1_1_1_1_2_7"/><span class="lab"><font color="Green" title="E82">A63</font></span>: 
<font color="Maroon" title="c19">i9</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">yq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="txt" href="#E33:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="txt" href="#E1:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p4">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E8:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E3:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E74">A53</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p4">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E7:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E9:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="txt" href="#E8:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E77">A56</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E3:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E74">A53</font></span></a>, <a class="txt" href="#E14:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E82">A63</font></span></a>, <a class="txt" href="#E13:43_2_1_1_1_1_2_7"><span class="lab"><font color="Green" title="E81">A62</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E35:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A64</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p2">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E14:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E36:43_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A65</font></span>: 
<font color="Maroon" title="c16">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="c9">yq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E16:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E37:43_2_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="txt" href="#E32:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><a NAME="E38:43_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A66</font></span>: 
<font color="Maroon" title="c15">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="c9">yq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="txt" href="#E30:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E39:43_2_1_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">ypq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">sqk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E10:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E31:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c10">ypq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p4">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E19:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="txt" href="#E24:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E63">A42</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E35:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="txt" href="#E23:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A41</font></span></a>, <a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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="c5">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p5">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="pre_poly.html#K7" title="PRE_POLY:func.7">SgmX</a> (<span class="p0">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a></span>)</span>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A64</font></span></a>, <a class="txt" href="#E28:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E66">A45</font></span></a>, <a class="ref" href="vectsp_1.html#D7" title="VECTSP_1:def.7">VECTSP_1:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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="c9">yq</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E19:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="txt" href="#E24:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="txt" href="#E36:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E73">A65</font></span></a></span>
;<br/><span class="kw">hence </span> <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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#K4" title="RLVECT_1:func.4">Sum</a> <font color="Maroon" title="c9">yq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E8:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E38:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E74">A66</font></span></a>, <a class="txt" href="#E34:43_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A50</font></span></a>, <a class="ref" href="polynom2.html#T5" target="_self" title="POLYNOM2:th.5">Th7</a></span>
<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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="polynom2.html#T19" target="_self" title="POLYNOM2:th.19">Th21</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:43_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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2"><span class="kw">case </span></a><a NAME="E1:43_2_1_2"/><span class="lab"><font color="Green" title="E48">A67</font></span>: 
 not <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c5">q</font>,<font color="Maroon" title="c6">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="c6">x</font>)</span>)</span></span><br/><div class="add"><span class="lab"><font color="Green" title="E49">A68</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom1.html#T15" title="POLYNOM1:th.15">POLYNOM1:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:43_2_1_2"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c3">p</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D4" title="RLVECT_1:def.4">RLVECT_1:def 4</a></span>
;<br/><a NAME="E3:43_2_1_2"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
 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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span> implies <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_2_1_2_2"/><span class="lab"><font color="Green" title="E51">A70</font></span>: 
<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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:43_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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_2_2"><span class="lab"><font color="Green" title="E51">A70</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/><div class="add"><a NAME="E2:43_2_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E2:43_2_1_2"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E52">A71</font></span>: 
<font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c11">u</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">n</font> ;<br/><a NAME="E3:43_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E53">A72</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>;<br/><a NAME="E4:43_2_1_2_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c12">u</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:43_2_1_2"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E1:43_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_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#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E52">A71</font></span></a>, <a class="txt" href="#E3:43_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E53">A72</font></span></a>, <a class="ref" href="polynom1.html#D3" title="POLYNOM1:def.3">POLYNOM1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E4:43_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>  <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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#K2" title="POLYNOM1:func.2">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a></span>;<br/><a NAME="E5:43_2_1_2"/><span class="kw">then </span>
 <a href="polynom1.html#K2" title="POLYNOM1:func.2">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="polynom1.html#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">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="c4">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#K2" title="POLYNOM1:func.2">Support</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E3:43_2_1_2"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><span class="kw">hence </span> <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">x</font>) = 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="polynom1.html#K3" title="POLYNOM1:func.3">.</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> (<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">x</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E1:43_2_1_2"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="polynom2.html#E36"><span class="lab"><font color="Green" title="E34">Lm10</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> (<font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">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="c6">x</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:43_2_1_2"><span class="lab"><font color="Green" title="E49">A68</font></span></a>, <a class="ref" href="polynom2.html#T19" target="_self" title="POLYNOM2:th.19">Th21</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:43"/>
 <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#K5" title="POLYNOM1:func.5">+</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">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="c6">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="c5">q</font>,<font color="Maroon" title="c6">x</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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