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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#V2">admissible</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_2.html#V2">domRing-like</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="polynom7.html#V1">non-zero</a> <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>











<b>assume </b><a NAME="E1:50"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="groeb_3.html#T10" target="_self">Th10</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="groeb_3.html#K6" target="_self">Low</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>;<br/>
<a NAME="E3:50"/><i><font color="Green">E45</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E4:50"/><i><font color="Green">E46</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K3" target="_self">Upper_Support</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E5:50"/><i><font color="Green">E47</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K3" target="_self">Upper_Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> &amp;  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="groeb_3.html#K4" target="_self">Lower_Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E6:50"/><i><font color="Green">E48</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E7:50"/><i><font color="Green">E49</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E5:50"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:50"/><i><font color="Green">E50</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E4:50"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E9:50"/><i><font color="Green">E51</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#T26" target="_self">Th26</a></i>;<br/>
<a NAME="E10:50"/><i><font color="Green">E52</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:50_1"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>, <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="groeb_3.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_3.html#E8:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E3:50_1"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:50"/><i><font color="Green">E53</font></i>: 
(  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</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">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:50_2"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>, <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="txt" href="groeb_3.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="txt" href="groeb_3.html#E8:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_1.html#T21">CARD_1:21</a></i>;<br/>
<b>hence </b><a NAME="E3:50_2"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T46">CARD_1:46</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E54</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">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_3"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50"><i><font color="Green">E45</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self">Th8</a></i>;<br/><div><i><font color="Green">E57</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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">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_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_3.html#E3:50"><i><font color="Green">E45</font></i></a>, <a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self">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">E58</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><div><i><font color="Green">E59</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">c<sub>10</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:50_3_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_3_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E6:50"><i><font color="Green">E48</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:50_3_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R2">&lt;</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#T20" target="_self">Th20</a>, <a class="txt" href="groeb_3.html#E1:50_3"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_1"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E4:50_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R2">&lt;</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T4" target="_self">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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D3">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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_3_2_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green">E47</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">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">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> </span>)</span>;<br/><a NAME="E3:50_3_2"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_3_2_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font>
;<br/>

<font color="Maroon">c<sub>11</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font>] )
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_3_2_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_3_2"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>, <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:50_3_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="groeb_3.html#E3:50_3_2"><i><font color="Green">E60</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">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_3_2_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><a NAME="E2:50_3_2_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/><i><font color="Green">E63</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/><font color="Maroon">c<sub>13</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/><a NAME="E6:50_3_2_3"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:50_3_2_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_3_2"><i><font color="Green">E61</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E6:50_3_2_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_3_2_3"><i><font color="Green">E62</font></i></a>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
;<br/></div><b>end;</b></div><a NAME="E7:50_3_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:50_3_2"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E10:50"><i><font color="Green">E52</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:50_3_2"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:50_3_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/><b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E4:50"><i><font color="Green">E46</font></i></a>, <a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">Def2</a></i>
;<br/><a NAME="E11:50_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_3_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="groeb_3.html#E7:50"><i><font color="Green">E49</font></i></a>, <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:50_3_2"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green">E63</font></i></a></i>;<br/><font color="Maroon">c<sub>12</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/><a NAME="E16:50_3_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_3_2"><i><font color="Green">E61</font></i></a></i>;<br/><font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E16:50_3_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="polynom1.html#T52">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">E57</font></i></a>, <a class="txt" href="groeb_3.html#E13:50_3_2"><i><font color="Green">E63</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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_3.html#T28" target="_self">Th28</a>, <a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_3"><i><font color="Green">E56</font></i></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">E55</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_4"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E3:50_4"/><i><font color="Green">E57</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="groeb_3.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E4:50_4"/>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_4"><i><font color="Green">E56</font></i></a>, <a class="ref" href="groeb_3.html#T9" target="_self">Th9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:50_4"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="polynom1.html#K10">-'</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E6:50_4"><i><font color="Green">E58</font></i></a>, <a class="ref" href="polynom1.html#T52">POLYNOM1:52</a></i>
;<br/><div><i><font color="Green">E60</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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_4_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">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#E1:50_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_3.html#E9:50"><i><font color="Green">E51</font></i></a>, <a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#T28" target="_self">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">E61</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><div><i><font color="Green">E62</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">c<sub>11</sub></font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><a NAME="E1:50_4_3_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><a NAME="E2:50_4_3_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green">E51</font></i></a>, <a class="ref" href="groeb_3.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:50_4_3_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R2">&lt;</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#T20" target="_self">Th20</a>, <a class="txt" href="groeb_3.html#E1:50_4"><i><font color="Green">E55</font></i></a>, <a class="txt" href="groeb_3.html#E5:50"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_1"><i><font color="Green">E63</font></i></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">c<sub>11</sub></font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:50_4_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>11</sub></font> <a href="termord.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="groeb_3.html#T2" target="_self">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">E65</font></i></a>, <a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/></div><b>end;</b></div><a NAME="E5:50_4_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R2">&lt;</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#T5">TERMORD:5</a></i>;<br/><a NAME="E6:50_4_3_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="termord.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="termord.html#D3">TERMORD:def 3</a></i>;<br/><b>hence </b><a NAME="E7:50_4_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:50_4_3_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="groeb_3.html#E4:50"><i><font color="Green">E46</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">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">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">a<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> </span>)</span>;<br/><a NAME="E3:50_4_3"/><i><font color="Green">E63</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:50_4_3_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:50_4_3_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="groeb_3.html#E9:50"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:50_4_3_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:50_4"><i><font color="Green">E57</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font>
;<br/>

<font color="Maroon">c<sub>12</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:50_4_3_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>] )
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:50_4_3_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_3.html#E2:50_4_3"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>, <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:50_4_3"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>(<a class="txt" href="groeb_3.html#E3:50_4_3"><i><font color="Green">E63</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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:50_4_3_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E2:50_4_3_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E3:50_4_3_3"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/><i><font color="Green">E66</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/><font color="Maroon">c<sub>15</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">POLYNOM2:def 3</a></i>
;<br/><b>hence </b><font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="polynom7.html#K2">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>14</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:50_4_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green">E64</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>13</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="groeb_3.html#E1:50_4_3_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_4_3_3"><i><font color="Green">E66</font></i></a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E7:50_4_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:50_4_3"/>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E11:50"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E9:50_4_3"/><b>then </b>
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:50_4_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="card_1.html#T60">CARD_1:60</a></i>;<br/><b>then </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p3">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T21">CARD_1:21</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E5:50"><i><font color="Green">E47</font></i></a>, <a class="txt" href="groeb_3.html#E2:50"><i><font color="Green">E44</font></i></a>, <a class="ref" href="groeb_3.html#D2" target="_self">Def2</a></i>
;<br/><a NAME="E11:50_4_3"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <span class="p2">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:50_4_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="groeb_3.html#E8:50"><i><font color="Green">E50</font></i></a>, <a class="ref" href="triang_1.html#T3">TRIANG_1:3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:50_4_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E14:50_4_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E13:50_4_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E15:50_4_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K5" target="_self">Up</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font> ;<br/><font color="Maroon">c<sub>13</sub></font> = 
 <a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#D1">FUNCT_7:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_7.html#K1">In</a> <font color="Maroon">c<sub>13</sub></font>,<span class="p2">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="polynom2.html#K4">@</a> 
<b>by </b><i><a class="ref" href="polynom2.html#D3">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">E60</font></i></a>, <a class="txt" href="groeb_3.html#E13:50_4_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="groeb_3.html#E5:50_4_3"><i><font color="Green">E64</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:50_4"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="groeb_3.html#T28" target="_self">Th28</a>, <a class="txt" href="groeb_3.html#E1:50"><i><font color="Green">E43</font></i></a>, <a class="txt" href="groeb_3.html#E7:50_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="groeb_3.html#E6:50_4"><i><font color="Green">E58</font></i></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">term</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="polynom1.html#K9">+</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> iff <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><a href="groeb_3.html#K6" target="_self">Low</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E12:50"><i><font color="Green">E54</font></i></a></i>;<br/>


</div>
