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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L2">ZeroStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="polynom1.html#NM3">Series</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>;<br/>
<a NAME="E1:22"/><i><font color="Green">E19</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/>
<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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:22_1"/><b>then </b><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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/><a NAME="E4:22_1"/>
not <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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:22_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E5:22_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E6:22_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:22_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><b>hence </b><a NAME="E7:22_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"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E21</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"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</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/><a NAME="E3:22_2"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:22_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E4:22_2"/><i><font color="Green">E23</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><div><i><font color="Green">E24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_4.html#K1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="groeb_3.html#E1:22_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="groeb_3.html#E4:22_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:22_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:22_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/></div><b>end;</b></div><a NAME="E6:22_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E3:22_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/><a NAME="E7:22_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="polynom1.html#D9">POLYNOM1:def 9</a></i>;<br/><a NAME="E8:22_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:22_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="groeb_3.html#E5:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="groeb_3.html#E7:22_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E9:22_2"/>
<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> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E7:22_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/><i><font color="Green">E21</font></i>: 
 <a href="polynom1.html#K12">Support</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="groeb_3.html#E2:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3"><b>now </b></a><div class="add"><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>assume </b><a NAME="E1:22_3"/>
<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>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/><a NAME="E2:22_3"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="groeb_3.html#E4:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:22_3"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xboole_0.html#K4">\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="groeb_3.html#E1:22"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:22_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> 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"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="groeb_3.html#K2" target="_self">|</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="polynom1.html#K15">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div>
