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<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>



<a NAME="E1:25"/><i><font color="Green">E30</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/>
<a NAME="E2:25"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> is <a href="wellord1.html#V2">well-ordering</a>
 
<b>by </b><i><a class="ref" href="wellord2.html#T7">WELLORD2:7</a></i>;<br/>
<a NAME="E3:25"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> is <a href="orders_1.html#V3">being_linear-order</a>
 
<b>by </b><i><a class="ref" href="orders_1.html#T107">ORDERS_1:107</a></i>;<br/>
<a NAME="E4:25"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/>
<a NAME="E5:25"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green">E36</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E38</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E48</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:25_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom2.html#K2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:25_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:25_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E2:25_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:25_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E4:25_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:25_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E7:25_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i>, , , <a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/><a NAME="E9:25_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E10:25_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E11:25_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="bagorder.html#T1" target="_self">Th1</a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/></div><b>end;</b></div><a NAME="E7:25_1"/><b>then </b>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T17">FUNCOP_1:17</a></i>;<br/><b>hence </b><a NAME="E8:25_1"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E8:25"/>
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E9:25"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">b</font> <a href="pboole.html#R6">=</a> <font color="Maroon">n</font> <a href="pre_circ.html#K2">--&gt;</a> 0
 
<b>by </b><i><a class="ref" href="polynom1.html#D15">POLYNOM1:def 15</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E11:25"/><i><font color="Green">E70</font></i>: 
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E12:25"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom2.html#K2">*</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_2"/>
 <a href="polynom2.html#K1">support</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:25_2"/>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E5:25_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:25"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E15:25"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<a NAME="E16:25"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom2.html#K2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>;<br/>
<a NAME="E17:25"/><b>then </b>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="polynom2.html#K2">*</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">b</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E18:25"/>
 <a href="bagorder.html#K3" target="_self">TotDegree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="relat_1.html#T64">RELAT_1:64</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>


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