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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c2">T</font> be  <a href="relat_2.html#V6" title="RELAT_2:attr.6">connected</a> <a href="bagorder.html#V2" title="BAGORDER:attr.2">admissible</a> <a href="bagorder.html#NM1" title="BAGORDER:NM.1">TermOrder</a> of <font color="Maroon" title="c1">n</font>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V2" title="GROUP_1:attr.2">unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_2.html#V1" title="VECTSP_2:attr.1">domRing-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="polynom1.html#NM4" title="POLYNOM1:NM.4">Polynomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/><b>let </b><font color="Maroon" title="c5">m</font> be  <a href="polynom7.html#V1" title="POLYNOM7:attr.1">non-zero</a> <a href="polynom7.html#NM1" title="POLYNOM7:NM.1">Monomial</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">L</font>;<br/><b>let </b><font color="Maroon" title="c6">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>











<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E41">A1</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font></span>)</span>
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T10" target="_self" title="GROEB_3:th.10">Th10</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/>

<b>set </b><font color="Maroon" title="c8">l</font> =  <a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font>;<br/>
<b>set </b><font color="Maroon" title="c9">lm</font> =  <a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font>;<br/>
<a NAME="E3:50"/><i><font color="Green" title="E43">A3</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self" title="GROEB_3:th.26">Th26</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E44">A4</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_3.html#K3" title="GROEB_3:func.3">Upper_Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font> &amp;  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_3.html#K4" title="GROEB_3:func.4">Lower_Support</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E32"><i><font color="Green" title="E25">Lm3</font></i></a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green" title="E45">A5</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_3.html#K3" title="GROEB_3:func.3">Upper_Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font> &amp;  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="groeb_3.html#K4" title="GROEB_3:func.4">Lower_Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E32"><i><font color="Green" title="E25">Lm3</font></i></a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green" title="E46">A6</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self" title="GROEB_3:th.26">Th26</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green" title="E47">A7</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green" title="E48">A8</font></i>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green" title="E49">A9</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> &amp;  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self" title="GROEB_3:th.26">Th26</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green" title="E50">A10</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="groeb_3.html#E7:50"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="groeb_3.html#E8:50"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46" title="CARD_1:th.46">CARD_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:50"/><i><font color="Green" title="E51">A11</font></i>: 
(  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_2"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:50_2"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="groeb_3.html#E7:50"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="groeb_3.html#E8:50"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E3:50_2"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46" title="CARD_1:th.46">CARD_1:46</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E52">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3"/><i><font color="Green" title="E53">A13</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_3"/><b>then </b><i><font color="Green" title="E54">A14</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self" title="GROEB_3:th.8">Th8</a></i>;<br/><div><i><font color="Green" title="E55">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_3_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_3"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:50_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E3:50"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3"><i><font color="Green" title="E53">A13</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self" title="GROEB_3:th.28">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_3_2"/><i><font color="Green" title="E56">A16</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><div><i><font color="Green" title="E57">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">b'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:50_3_2_1"/><i><font color="Green" title="E58">A18</font></i>: 
<font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_3_2_1"/><b>then </b><i><font color="Green" title="E59">A19</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E6:50"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self" title="GROEB_3:th.8">Th8</a></i>;<br/><a NAME="E3:50_3_2_1"/>
<font color="Maroon" title="c7">b</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3"><i><font color="Green" title="E53">A13</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_1"><i><font color="Green" title="E58">A18</font></i></a>, <a class="ref" href="groeb_3.html#T20" target="_self" title="GROEB_3:th.20">Th20</a></i>;<br/><a NAME="E4:50_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T4" target="_self" title="GROEB_3:th.4">Th4</a></i>;<br/><a NAME="E5:50_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c10">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/><b>hence </b><a NAME="E6:50_3_2_1"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c10">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2"><i><font color="Green" title="E56">A16</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_3_2_1"><i><font color="Green" title="E59">A19</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> $1,<span class="p3">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>;<br/><a NAME="E3:50_3_2"/><i><font color="Green" title="E58">A20</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_3_2_2"/><i><font color="Green" title="E59">A21</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c11">x'</font> = <font color="Maroon" title="c10">x</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">x'</font>
;<br/>

<font color="Maroon" title="c11">x'</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c11">x'</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c11">x'</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E4:50_3_2_2"/>
( <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">x'</font>] )
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_3_2"><i><font color="Green" title="E57">A17</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_2"><i><font color="Green" title="E59">A21</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c10">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>, <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span><b> such that </b><br/><a NAME="E5:50_3_2"/><i><font color="Green" title="E59">A22</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="groeb_3.html#E3:50_3_2"><i><font color="Green" title="E58">A20</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:50_3_2_3"/><i><font color="Green" title="E60">A23</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x2</font> )
 ;<br/><a NAME="E2:50_3_2_3"/><b>then </b>
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">x</font> = <font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c14">y</font> = <font color="Maroon" title="c12">x2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><i><font color="Green" title="E61">A24</font></i>: <font color="Maroon" title="c14">y</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c14">y</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c14">y</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><font color="Maroon" title="c13">x</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c13">x</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c13">x</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><a NAME="E6:50_3_2_3"/><b>then </b><i><font color="Green" title="E62">A25</font></i>: 
( <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">x</font> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:50_3_2"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_3"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="groeb_3.html#E4:50_3_2_3"><i><font color="Green" title="E61">A24</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c11">x1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c13">x</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon" title="c12">x2</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50_3_2_3"><i><font color="Green" title="E60">A23</font></i></a>, <a class="txt" href="groeb_3.html#E6:50_3_2_3"><i><font color="Green" title="E62">A25</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
;<br/></div><b>end;</b></div><a NAME="E7:50_3_2"/><b>then </b><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c10">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:50_3_2"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E10:50"><i><font color="Green" title="E50">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:50_3_2"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:50_3_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></i>;<br/><b>then </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>
<br/>.= 
<font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>
;<br/><a NAME="E11:50_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:50"><i><font color="Green" title="E47">A7</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2"><i><font color="Green" title="E56">A16</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">bb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:50_3_2"/><i><font color="Green" title="E61">A27</font></i>: 
( <font color="Maroon" title="c11">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c11">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">bb</font> = <font color="Maroon" title="c11">bb</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green" title="E61">A27</font></i></a></i>;<br/><font color="Maroon" title="c12">bb</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c12">bb</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c12">bb</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><a NAME="E16:50_3_2"/><b>then </b><i><font color="Green" title="E62">A28</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E5:50_3_2"><i><font color="Green" title="E59">A22</font></i></a>, <a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green" title="E61">A27</font></i></a></i>;<br/><font color="Maroon" title="c12">bb</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">bb</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E16:50_3_2"><i><font color="Green" title="E62">A28</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
;<br/><b>hence </b><a NAME="E18:50_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50_3"><i><font color="Green" title="E55">A15</font></i></a>, <a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green" title="E61">A27</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:50_3"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_3"><i><font color="Green" title="E54">A14</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self" title="GROEB_3:th.28">Th28</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_4"/><i><font color="Green" title="E53">A29</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_4"/><b>then </b><i><font color="Green" title="E54">A30</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/><a NAME="E3:50_4"/><i><font color="Green" title="E55">A31</font></i>: 
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span> <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"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self" title="GROEB_3:th.9">Th9</a></i>;<br/><a NAME="E4:50_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Olive" title="b1">u</font></span>)</span> where <font color="Olive" title="b1">u</font> is    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_4"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="groeb_3.html#T9" target="_self" title="GROEB_3:th.9">Th9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">u</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E6:50_4"/><i><font color="Green" title="E56">A32</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c10">u</font> &amp; <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> )
 ;<br/><i><font color="Green" title="E57">A33</font></i>: <font color="Maroon" title="c7">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="polynom1.html#K10" title="POLYNOM1:func.10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon" title="c10">u</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E6:50_4"><i><font color="Green" title="E56">A32</font></i></a>, <a class="ref" href="polynom1.html#T52" title="POLYNOM1:th.52">POLYNOM1:52</a></i>
;<br/><div><i><font color="Green" title="E58">A34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_4_2"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_4"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:50_4_2"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E9:50"><i><font color="Green" title="E49">A9</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self" title="GROEB_3:th.28">Th28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_4_3"/><i><font color="Green" title="E59">A35</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><div><i><font color="Green" title="E60">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">b'</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">n</font>;<br/><b>assume </b><a NAME="E1:50_4_3_1"/><i><font color="Green" title="E61">A37</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><a NAME="E2:50_4_3_1"/><b>then </b><i><font color="Green" title="E62">A38</font></i>: 
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self" title="GROEB_3:th.8">Th8</a></i>;<br/><a NAME="E3:50_4_3_1"/><i><font color="Green" title="E63">A39</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green" title="E45">A5</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4"><i><font color="Green" title="E53">A29</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_1"><i><font color="Green" title="E61">A37</font></i></a>, <a class="ref" href="groeb_3.html#T20" target="_self" title="GROEB_3:th.20">Th20</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:50_4_3_1_1"/>
<font color="Maroon" title="c11">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">T</font>
 ;<br/><a NAME="E2:50_4_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c11">b'</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T2" target="_self" title="GROEB_3:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E3:50_4_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50_4_3_1"><i><font color="Green" title="E63">A39</font></i></a>, <a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><a NAME="E5:50_4_3_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="termord.html#R2" title="TERMORD:pred.2">&lt;</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#T5" title="TERMORD:th.5">TERMORD:5</a></i>;<br/><a NAME="E6:50_4_3_1"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="termord.html#R1" title="TERMORD:pred.1">&lt;=</a> <font color="Maroon" title="c11">b'</font>,<font color="Maroon" title="c2">T</font>
 <b>by </b><i><a class="ref" href="termord.html#D3" title="TERMORD:def.3">TERMORD:def 3</a></i>;<br/><b>hence </b><a NAME="E7:50_4_3_1"/>
<font color="Maroon" title="c11">b'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3"><i><font color="Green" title="E59">A35</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_4_3_1"><i><font color="Green" title="E62">A38</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> $2,<span class="p3">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>;<br/><a NAME="E3:50_4_3"/><i><font color="Green" title="E61">A40</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:50_4_3_2"/><i><font color="Green" title="E62">A41</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/>

<a NAME="E2:50_4_3_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green" title="E49">A9</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">x'</font> being    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E4:50_4_3_2"/><i><font color="Green" title="E63">A42</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c12">x'</font> &amp; <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50_4"><i><font color="Green" title="E55">A31</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">x'</font>
;<br/>

<font color="Maroon" title="c12">x'</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c12">x'</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c12">x'</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:50_4_3_2"/>
( <font color="Maroon" title="c12">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">x'</font>] )
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_4_3"><i><font color="Green" title="E60">A36</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_2"><i><font color="Green" title="E62">A41</font></i></a>, <a class="txt" href="groeb_3.html#E4:50_4_3_2"><i><font color="Green" title="E63">A42</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c11">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>, <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span><b> such that </b><br/><a NAME="E5:50_4_3"/><i><font color="Green" title="E62">A43</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="groeb_3.html#E3:50_4_3"><i><font color="Green" title="E61">A40</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:50_4_3_3"/><i><font color="Green" title="E63">A44</font></i>: 
( <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> )
 ;<br/><a NAME="E2:50_4_3_3"/><b>then </b>
( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E3:50_4_3_3"/><b>then </b>
( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c14">x1'</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c15">x2'</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><i><font color="Green" title="E64">A45</font></i>: <font color="Maroon" title="c14">x1'</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c14">x1'</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c14">x1'</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><font color="Maroon" title="c15">x2'</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c15">x2'</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c15">x2'</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><b>hence </b><font color="Maroon" title="c12">x1</font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c14">x1'</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green" title="E62">A43</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_3"><i><font color="Green" title="E63">A44</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c13">x2</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green" title="E62">A43</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_3"><i><font color="Green" title="E63">A44</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_4_3_3"><i><font color="Green" title="E64">A45</font></i></a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E7:50_4_3"/><b>then </b><i><font color="Green" title="E63">A46</font></i>: 
<font color="Maroon" title="c11">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:50_4_3"/>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E11:50"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:50_4_3"/><b>then </b>
 <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<font color="Maroon" title="c11">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:50_4_3"><i><font color="Green" title="E63">A46</font></i></a>, <a class="ref" href="card_1.html#T60" title="CARD_1:th.60">CARD_1:60</a></i>;<br/><b>then </b> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p3">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>
<br/>.= 
<font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green" title="E42">A2</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self" title="GROEB_3:def.2">Def2</a></i>
;<br/><a NAME="E11:50_4_3"/><b>then </b>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E8:50"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3"><i><font color="Green" title="E59">A35</font></i></a>, <a class="ref" href="triang_1.html#T3" title="TRIANG_1:th.3">TRIANG_1:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">bb</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:50_4_3"/><i><font color="Green" title="E64">A47</font></i>: 
( <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> &amp; <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> &amp; <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><a NAME="E14:50_4_3"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E13:50_4_3"><i><font color="Green" title="E64">A47</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><a NAME="E15:50_4_3"/><b>then </b>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" title="GROEB_3:func.5">Up</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c13">bb'</font> = <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">bb</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font> ;<br/><font color="Maroon" title="c13">bb'</font> = 
 <a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c13">bb'</font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1" title="FUNCT_7:func.1">In</a> <font color="Maroon" title="c13">bb'</font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3" title="POLYNOM2:def.3">POLYNOM2:def 3</a></i>
;<br/><b>hence </b><a NAME="E18:50_4_3"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E8:50_4"><i><font color="Green" title="E58">A34</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green" title="E62">A43</font></i></a>, <a class="txt" href="groeb_3.html#E13:50_4_3"><i><font color="Green" title="E64">A47</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:50_4"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="groeb_3.html#E6:50_4"><i><font color="Green" title="E56">A32</font></i></a>, <a class="txt" href="groeb_3.html#E7:50_4"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self" title="GROEB_3:th.28">Th28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:50"/>
( <span class="p1">(<span class="default"><a href="polynom7.html#K2" title="POLYNOM7:func.2">term</a> <font color="Maroon" title="c5">m</font></span>)</span> <a href="polynom1.html#K9" title="POLYNOM1:func.9">+</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="polynom1.html#K28" title="POLYNOM1:func.28">*'</a> <font color="Maroon" title="c4">p</font></span>)</span>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> iff <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" title="GROEB_3:func.6">Low</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">T</font>,<font color="Maroon" title="c6">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E12:50"><i><font color="Green" title="E52">A12</font></i></a></i>;<br/>


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