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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">a<sub>1</sub></font> is <a href="wellord1.html#V2">well-ordering</a>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/><b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E2:42_1"/><i><font color="Green">E28</font></i>: 
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E4:42_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="partit_2.html#T9">PARTIT_2:9</a></i>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1"/>
not  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellord1.html#V1">well_founded</a>
 ;<br/><a NAME="E2:42_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
not  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="wellord1.html#T5">WELLORD1:5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/><a NAME="E3:42_1_1"/>
not  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="normsp_1.html#NM2">sequence</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)<b> such that </b><br/><a NAME="E5:42_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="wellfnd1.html#V3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> 0 as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><a NAME="E7:42_1_1"/><i><font color="Green">E33</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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><a NAME="E8:42_1_1"/>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></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="E9:42_1_1"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></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="E10:42_1_1"/><b>then </b>
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:42_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="orders_1.html#T133">ORDERS_1:133</a></i>;<br/><a NAME="E11:42_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font> <a href="orders_1.html#R3">linearly_orders</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T134">ORDERS_1:134</a></i>;<br/><a NAME="E12:42_1_1"/><b>then </b><i><font color="Green">E35</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_1"/>
 <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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="bagorder.html#T20" target="_self">Th20</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:42_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><a NAME="E5:42_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:42_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E9:42_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E10:42_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="polynom1.html#T56">POLYNOM1:56</a></i>;<br/><b>hence </b><a NAME="E11:42_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E14:42_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><a NAME="E15:42_1_1"/><b>then </b><i><font color="Green">E37</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <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 <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> );<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E2:42_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E19:42_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E17:42_1_1"><i><font color="Green">E38</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0;<br/><a NAME="E20:42_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:42_1_1_3"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><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="E1:42_1_1_3_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</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> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<b>assume </b><a NAME="E2:42_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:42_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:42_1_1_3_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>12</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> &amp; <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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <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>10</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_2.html#T11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E6:42_1_1_3_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_3_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> or <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E6:42_1_1_3_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:42_1_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E11:42_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="bagorder.html#E14:42_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/><a NAME="E3:42_1_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E4:42_1_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="yellow_1.html#K1">RelIncl</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E14:42_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><a NAME="E5:42_1_1_3_1_1_2"/><b>then </b>
<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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1">support</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="wellord2.html#D1">WELLORD2:def 1</a></i>;<br/><b>hence </b><a NAME="E6:42_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:42_1_1_3"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:42_1_1_3_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">c<sub>12</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="E2:42_1_1_3_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E4:42_1_1_3_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:42_1_1_3_2"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E7:42_1_1_3_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_3_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E9:42_1_1_3_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="bagorder.html#D8" target="_self">Def8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_3_2_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_3_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E9:42_1_1_3_2"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_3_2_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1_1_3_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_3_2"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_3_2_1_3"><b>suppose </b></a><a NAME="E1:42_1_1_3_2_1_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:42_1_1_3_2_1_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E3:42_1_1_3_2_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_3_2_1_3"><i><font color="Green">E49</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E4:42_1_1_3_2_1_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bagorder.html#E7:42_1_1_3_2"><i><font color="Green">E46</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_1_1_3_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E8:42_1_1_3_2"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:42_1_1_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E1:42_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3"><i><font color="Green">E42</font></i></a>);</i><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <a href="dickson.html#K11">OrderedNAT</a>  <b>by </b><i><a class="ref" href="dickson.html#D15">DICKSON:def 15</a></i>;<br/><a NAME="E22:42_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>10</sub></font> is <a href="bagorder.html#V1" target="_self">non-increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="bagorder.html#D3" target="_self">BAGORDER:def 3</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="dickson.html#D15">DICKSON:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:42_1_1_4"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/>
<a NAME="E4:42_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:42_1_1_4"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E7:42_1_1_4"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E8:42_1_1_4"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:42_1_1_4"><i><font color="Green">E42</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E9:42_1_1_4"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="bagorder.html#E7:42_1_1_4"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>17</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:42_1_1_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:42_1_1_4"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_4_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:42_1_1_4"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bagorder.html#E11:42_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E13:42_1_1_4"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bagorder.html#E6:42_1_1_4"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bagorder.html#E9:42_1_1_4"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:42_1_1_4_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_4"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E11:42_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E13:42_1_1_4"><i><font color="Green">E48</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E15:42_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="dickson.html#K11">OrderedNAT</a> 
 <b>by </b><i><a class="ref" href="dickson.html#D14">DICKSON:def 14</a>, <a class="ref" href="dickson.html#D15">DICKSON:def 15</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> =  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>13</sub></font> =  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> #);<br/><a NAME="E23:42_1_1"/>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_2.html#R7">is_strongly_connected_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E24:42_1_1"/><b>then </b>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E25:42_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="partit_2.html#T9">PARTIT_2:9</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E27:42_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:42_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="bagorder.html#T11" target="_self">Th11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ex <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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font> );<br/><div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="bagorder.html#T21" target="_self">Th21</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font> as    <a href="polynom1.html#M1">Element</a> of  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>thus </b><a NAME="E4:42_1_1_5"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>19</sub></font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font><b> such that </b><br/><a NAME="E30:42_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E28:42_1_1"><i><font color="Green">E44</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> #) ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:42_1_1_6"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E30:42_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:42_1_1_6"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E30:42_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:42_1_1_6"/>
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E6:42_1_1_6"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:42_1_1_6"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:42_1_1_6"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green">E39</font></i></a></i>;<br/><a NAME="E11:42_1_1_6"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E12:42_1_1_6"/><b>then </b><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E27:42_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><b>thus </b><a NAME="E13:42_1_1_6"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1_1_6_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>

<a NAME="E2:42_1_1_6_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>19</sub></font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_6_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:42_1_1_6_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is   <a href="ordinal1.html#NM2">Ordinal</a>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_6_1_1_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_6_1_1_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_6_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_6_1_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_1"><i><font color="Green">E52</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_6_1_1_1_2"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_6_1_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_6"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_1_1_1_3"><b>suppose </b></a><a NAME="E1:42_1_1_6_1_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_6_1_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_6_1_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:42_1_1_6_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E6:42_1_1_6"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:42_1_1_6"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>22</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E16:42_1_1_6"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E17:42_1_1_6"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E18:42_1_1_6"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E6:42_1_1_6"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self">Def8</a></i>;<br/><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1_6_2"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_6_2_1"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_2_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_6_2_1_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_6_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_6"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_6"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_6"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bagorder.html#E17:42_1_1_6"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_6_2_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_6_2_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>then </b><font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> = 
0
<b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E16:42_1_1_6"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E16:42_1_1_6"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_2_1_2"><i><font color="Green">E57</font></i></a></i>
;<br/><b>hence </b><a NAME="E3:42_1_1_6_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E17:42_1_1_6"><i><font color="Green">E53</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="relat_1.html#K7">|</a> <font color="Maroon">c<sub>8</sub></font> as   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="bagorder.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E21:42_1_1_6"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1_6"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>25</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:42_1_1_6_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E2:42_1_1_6_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><a NAME="E3:42_1_1_6_3"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E18:42_1_1_6"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green">E57</font></i></a></i>;<br/><b>thus </b><font color="Maroon">c<sub>23</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>25</sub></font> = 
<font color="Maroon">c<sub>18</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">c<sub>24</sub></font> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>25</sub></font>
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bagorder.html#E3:42_1_1_6_3"><i><font color="Green">E58</font></i></a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:42_1_1_6"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E17:42_1_1_6"><i><font color="Green">E53</font></i></a>, <a class="txt" href="bagorder.html#E19:42_1_1_6"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bagorder.html#E21:42_1_1_6"><i><font color="Green">E56</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self">Def8</a></i>;<br/></div><b>end;</b></div><a NAME="E33:42_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> is <a href="wellfnd1.html#V3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><a NAME="E34:42_1_1"/><b>then </b>
not  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> #) is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/><a NAME="E35:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> #)
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2">WELLFND1:def 2</a></i>;<br/><a NAME="E36:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E25:42_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><a NAME="E37:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>8</sub></font> is <a href="wellord1.html#V2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/><b>hence </b><a NAME="E38:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:42_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97">ORDERS_1:97</a></i>;<br/><a NAME="E7:42_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R1">is_reflexive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/><a NAME="E8:42_1"/><i><font color="Green">E33</font></i>: 
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:42_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/><a NAME="E9:42_1"/><i><font color="Green">E34</font></i>: 
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:42_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/><a NAME="E10:42_1"/><i><font color="Green">E35</font></i>: 
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="relat_2.html#R6">is_connected_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="orders_1.html#T92">ORDERS_1:92</a></i>;<br/><a NAME="E11:42_1"/>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R1">is_well_founded_in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="wellord1.html#T5">WELLORD1:5</a></i>;<br/><a NAME="E12:42_1"/><b>then </b>
 <a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#R2">well_orders</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" target="_self">InvLexOrder</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="bagorder.html#E7:42_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="bagorder.html#E9:42_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="wellord1.html#D5">WELLORD1:def 5</a></i>;<br/><b>hence </b><a NAME="E13:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="ref" href="wellord1.html#T8">WELLORD1:8</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:42"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="txt" href="bagorder.html#E1:42"><i><font color="Green">E26</font></i></a>);</i><br/>


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