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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> $1 is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>;<br/>
<div><i><font color="Green" title="E26">A1</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" title="c1">o</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E27">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 ;<br/><b>set </b><font color="Maroon" title="c2">ilo</font> =  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>;<br/><a NAME="E2:42_1"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92" title="ORDERS_1:th.92">ORDERS_1:92</a></i>;<br/><a NAME="E4:42_1"/><b>then </b><i><font color="Green" title="E29">A4</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T9" title="PARTIT_2:th.9">PARTIT_2:9</a></i>;<br/><div><i><font color="Green" title="E30">A5</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" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> is <a href="wellord1.html#V1" title="WELLORD1:attr.1">well_founded</a>
 ;<br/><a NAME="E2:42_1_1"/><b>then </b><i><font color="Green" title="E31">A6</font></i>: 
not  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="wellord1.html#T5" title="WELLORD1:th.5">WELLORD1:5</a></i>;<br/><b>set </b><font color="Maroon" title="c3">R</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #);<br/><b>set </b><font color="Maroon" title="c4">ir</font> = the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #);<br/><a NAME="E3:42_1_1"/>
not  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #) is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)<b> such that </b><br/><a NAME="E5:42_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>set </b><font color="Maroon" title="c7">s</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span>;<br/><a NAME="E7:42_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></i>;<br/><a NAME="E8:42_1_1"/>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord2.html#T7" title="WELLORD2:th.7">WELLORD2:7</a></i>;<br/><a NAME="E9:42_1_1"/><b>then </b>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font> <a href="orders_1.html#NR3" title="ORDERS_1:NR.3">is_linear-order</a> 
 <b>by </b><i><a class="ref" href="orders_1.html#T107" title="ORDERS_1:th.107">ORDERS_1:107</a></i>;<br/><a NAME="E10:42_1_1"/><b>then </b>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:42_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="orders_1.html#T133" title="ORDERS_1:th.133">ORDERS_1:133</a></i>;<br/><a NAME="E11:42_1_1"/><b>then </b><i><font color="Green" title="E34">A9</font></i>: 
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font> <a href="orders_1.html#R3" title="ORDERS_1:pred.3">linearly_orders</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T134" title="ORDERS_1:th.134">ORDERS_1:134</a></i>;<br/><a NAME="E12:42_1_1"/><b>then </b><i><font color="Green" title="E35">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">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" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E2:42_1_1_1"/><b>then </b><i><font color="Green" title="E36">A11</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="polynom1.html#K16" title="POLYNOM1:func.16">EmptyBag</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="bagorder.html#T20" target="_self" title="BAGORDER:th.20">Th20</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><a NAME="E4:42_1_1_1"/><i><font color="Green" title="E37">A12</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><a NAME="E5:42_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c6">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> #)
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:42_1_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <b>and </b><br/><a NAME="E8:42_1_1_1"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c8">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c6">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c9">i</font>
 <b>and </b><br/><a NAME="E9:42_1_1_1"/>
for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> holds <br/><font color="Maroon" title="c8">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1_1_1"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self" title="BAGORDER:def.8">Def8</a></i>;<br/><b>thus </b><a NAME="E10:42_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_1"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_1"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="polynom1.html#T56" title="POLYNOM1:th.56">POLYNOM1:56</a></i>;<br/></div><b>end;</b></div><a NAME="E14:42_1_1"/><b>then </b><i><font color="Green" title="E36">A14</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></i>;<br/><a NAME="E15:42_1_1"/><b>then </b><i><font color="Green" title="E37">A15</font></i>: 
<span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">j</font> = <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></span>)</span> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font> );<br/><div><i><font color="Green" title="E38">A16</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" title="c9">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">x</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font>;<br/><b>thus </b><a NAME="E2:42_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c9">t</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E19:42_1_1"/><i><font color="Green" title="E39">A17</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c9">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E17:42_1_1"><i><font color="Green" title="E38">A16</font></i></a>);<br/></i><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">i</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font>  st <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/><a NAME="E20:42_1_1"/><i><font color="Green" title="E40">A18</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</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" title="E41">A19</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<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" title="c10">i</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c11">b</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font>;<br/>



<b>assume </b><a NAME="E1:42_1_1_3_1"/><i><font color="Green" title="E42">A20</font></i>: 
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> )
 ;<br/>

<b>assume </b><a NAME="E2:42_1_1_3_1"/>
<font color="Maroon" title="c11">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E3:42_1_1_3_1"/><b>then </b>
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="ref" href="polynom1.html#D7" title="POLYNOM1:def.7">POLYNOM1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:42_1_1_3_1"/><i><font color="Green" title="E43">A21</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<a NAME="E6:42_1_1_3_1"/><i><font color="Green" title="E44">A22</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</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" title="E43">A21</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">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" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> or <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</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" title="E44">A22</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">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" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</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" title="E42">A20</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green" title="E43">A21</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" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E11:42_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="bagorder.html#E14:42_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="triang_1.html#D2" title="TRIANG_1:def.2">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" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><a NAME="E4:42_1_1_3_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E14:42_1_1"><i><font color="Green" title="E36">A14</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</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" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <font color="Maroon" title="c6">a</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" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_3_1"><i><font color="Green" title="E42">A20</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">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" title="E42">A20</font></i></a>, <a class="txt" href="bagorder.html#E5:42_1_1_3_1"><i><font color="Green" title="E43">A21</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">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" title="E42">A23</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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" title="c10">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:42_1_1_3_2"/><i><font color="Green" title="E43">A24</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font>  st <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b2">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c11">i</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c12">b1</font> be   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font>;<br/>



<b>assume </b><a NAME="E2:42_1_1_3_2"/><i><font color="Green" title="E44">A25</font></i>: 
( <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">n</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E4:42_1_1_3_2"/><i><font color="Green" title="E45">A26</font></i>: 
<font color="Maroon" title="c13">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green" title="E44">A25</font></i></a></i>;<br/>
<a NAME="E5:42_1_1_3_2"/>
( <font color="Maroon" title="c12">b1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">b1</font>,<font color="Maroon" title="c13">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> )
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">i1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E7:42_1_1_3_2"/><i><font color="Green" title="E46">A27</font></i>: 
<font color="Maroon" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <b>and </b><br/><a NAME="E8:42_1_1_3_2"/><i><font color="Green" title="E47">A28</font></i>: 
<font color="Maroon" title="c12">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c14">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c14">i1</font>
 <b>and </b><br/><a NAME="E9:42_1_1_3_2"/><i><font color="Green" title="E48">A29</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> holds <br/><font color="Maroon" title="c12">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="ref" href="bagorder.html#D8" target="_self" title="BAGORDER:def.8">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" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">i</font> or <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font> or <font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">i1</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">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" title="c14">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_3_2_1_1"/>
<font color="Maroon" title="c12">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_3_2"><i><font color="Green" title="E45">A26</font></i></a>, <a class="txt" href="bagorder.html#E9:42_1_1_3_2"><i><font color="Green" title="E48">A29</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" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_3_2_1_2"/>
<font color="Maroon" title="c12">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_3_2"><i><font color="Green" title="E47">A28</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" title="E49">A30</font></i>: 
<font color="Maroon" title="c11">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">i1</font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:42_1_1_3_2_1_3"/>
<font color="Maroon" title="c12">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:42_1_1_3_2_1_3"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">i1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_3_2"><i><font color="Green" title="E44">A25</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_3_2_1_3"><i><font color="Green" title="E49">A30</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E4:42_1_1_3_2_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_3_2"><i><font color="Green" title="E43">A24</font></i></a>, <a class="txt" href="bagorder.html#E7:42_1_1_3_2"><i><font color="Green" title="E46">A27</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_3_2"><i><font color="Green" title="E47">A28</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" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E1:42_1_1_3"><i><font color="Green" title="E41">A19</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_3"><i><font color="Green" title="E42">A23</font></i></a>);</i><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c10">t</font> = <font color="Maroon" title="c9">t</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a>  <b>by </b><i><a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a></i>;<br/><a NAME="E22:42_1_1"/><i><font color="Green" title="E41">A31</font></i>: 
<font color="Maroon" title="c10">t</font> is <a href="bagorder.html#V1" title="BAGORDER:attr.1">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" title="c11">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="bagorder.html#D3" target="_self" title="BAGORDER:def.3">BAGORDER:def 3</a><br/>

<b>reconsider </b><font color="Maroon" title="c12">tn</font> = <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c13">tn1</font> = <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">fn</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c15">fn1</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E3:42_1_1_4"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon" title="c15">fn1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">fn</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/>
<a NAME="E4:42_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">fn1</font>,<font color="Maroon" title="c14">fn</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c16">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:42_1_1_4"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <b>and </b><br/><a NAME="E7:42_1_1_4"/><i><font color="Green" title="E44">A34</font></i>: 
<font color="Maroon" title="c15">fn1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">fn</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c16">i</font>
 <b>and </b><br/><a NAME="E8:42_1_1_4"/><i><font color="Green" title="E45">A35</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> holds <br/><font color="Maroon" title="c15">fn1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fn</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:42_1_1_4"><i><font color="Green" title="E42">A32</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self" title="BAGORDER:def.8">Def8</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c17">bn</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E10:42_1_1_4"/><i><font color="Green" title="E46">A36</font></i>: 
( <font color="Maroon" title="c14">fn</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c17">bn</font> &amp; <font color="Maroon" title="c12">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">bn</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green" title="E39">A17</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">bn1</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E12:42_1_1_4"/><i><font color="Green" title="E47">A37</font></i>: 
( <font color="Maroon" title="c15">fn1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c18">bn1</font> &amp; <font color="Maroon" title="c13">tn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">bn1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green" title="E39">A17</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" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">i</font> or <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font> )
 </span><b>by </b><i><a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">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" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_1"/>
<font color="Maroon" title="c13">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">tn</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:42_1_1_4"><i><font color="Green" title="E44">A34</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_4"><i><font color="Green" title="E46">A36</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_4"><i><font color="Green" title="E47">A37</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" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_2"/>
<font color="Maroon" title="c13">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">tn</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="bagorder.html#E6:42_1_1_4"><i><font color="Green" title="E43">A33</font></i></a>, <a class="txt" href="bagorder.html#E7:42_1_1_4"><i><font color="Green" title="E44">A34</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" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_4_1_1_3"/>
<font color="Maroon" title="c13">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">tn</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_4"><i><font color="Green" title="E45">A35</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_4"><i><font color="Green" title="E46">A36</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_4"><i><font color="Green" title="E47">A37</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="E14:42_1_1_4"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="dickson.html#K11" title="DICKSON:func.11">OrderedNAT</a> 
 <b>by </b><i><a class="ref" href="dickson.html#D14" title="DICKSON:def.14">DICKSON:def 14</a>, <a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c8">j</font>;<br/><b>set </b><font color="Maroon" title="c12">iln</font> =  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font>;<br/><b>set </b><font color="Maroon" title="c13">S</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font></span>)</span> #);<br/><a NAME="E23:42_1_1"/>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font> <a href="relat_2.html#R7" title="RELAT_2:pred.7">is_strongly_connected_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="bagorder.html#D7" target="_self" title="BAGORDER:def.7">Def7</a></i>;<br/><a NAME="E24:42_1_1"/><b>then </b>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T92" title="ORDERS_1:th.92">ORDERS_1:92</a></i>;<br/><a NAME="E25:42_1_1"/><b>then </b><i><font color="Green" title="E42">A38</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="partit_2.html#T9" title="PARTIT_2:th.9">PARTIT_2:9</a></i>;<br/><b>consider </b><font color="Maroon" title="c14">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E27:42_1_1"/><i><font color="Green" title="E43">A39</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c14">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:42_1_1"><i><font color="Green" title="E41">A31</font></i></a>, <a class="ref" href="bagorder.html#T11" target="_self" title="BAGORDER:th.11">Th11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">b</font> being  <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> );<br/><div><i><font color="Green" title="E44">A40</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" title="c15">x</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">x</font></span>)</span> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c8">j</font> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="bagorder.html#T21" target="_self" title="BAGORDER:th.21">Th21</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">y</font> as    <a href="polynom1.html#M1" title="POLYNOM1:mode.1">Element</a> of  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font> <b>by </b><i><a class="ref" href="polynom1.html#D14" title="POLYNOM1:def.14">POLYNOM1:def 14</a></i>;<br/><b>take </b><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">y</font>;<br/><b>thus </b><a NAME="E4:42_1_1_5"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c19">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon" title="c15">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font><b> such that </b><br/><a NAME="E30:42_1_1"/><i><font color="Green" title="E45">A41</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c15">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E28:42_1_1"><i><font color="Green" title="E44">A40</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon" title="c16">g</font> = <font color="Maroon" title="c15">g</font> as   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</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" title="c17">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>consider </b><font color="Maroon" title="c18">b</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E2:42_1_1_6"/><i><font color="Green" title="E46">A42</font></i>: 
( <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> &amp; <font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E30:42_1_1"><i><font color="Green" title="E45">A41</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c19">b1</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E4:42_1_1_6"/><i><font color="Green" title="E47">A43</font></i>: 
( <font color="Maroon" title="c19">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E30:42_1_1"><i><font color="Green" title="E45">A41</font></i></a></i>;<br/><a NAME="E5:42_1_1_6"/>
<font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E6:42_1_1_6"/><b>then </b><i><font color="Green" title="E48">A44</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">b1</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>consider </b><font color="Maroon" title="c20">bb</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E8:42_1_1_6"/><i><font color="Green" title="E49">A45</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">bb</font> &amp; <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">bb</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green" title="E39">A17</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c21">bb1</font> being   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c1">o</font><b> such that </b><br/><a NAME="E10:42_1_1_6"/><i><font color="Green" title="E50">A46</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">bb1</font> &amp; <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">bb1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">j</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1"><i><font color="Green" title="E39">A17</font></i></a></i>;<br/><a NAME="E11:42_1_1_6"/>
( <font color="Maroon" title="c14">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font> &amp; <font color="Maroon" title="c14">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E12:42_1_1_6"/><b>then </b><i><font color="Green" title="E51">A47</font></i>: 
( <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p</font> &amp; <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">t</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E27:42_1_1"><i><font color="Green" title="E43">A39</font></i></a></i>;<br/><b>thus </b><a NAME="E13:42_1_1_6"/>
<font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">k</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" title="E52">A48</font></i>: 
not <font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">k</font>
 ;<br/>

<a NAME="E2:42_1_1_6_1"/><i><font color="Green" title="E53">A49</font></i>: 
( <font color="Maroon" title="c1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">b</font> &amp; <font color="Maroon" title="c1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c19">b1</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="c22">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:42_1_1_6_1_1"/><i><font color="Green" title="E54">A50</font></i>: 
<font color="Maroon" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 ;<br/><a NAME="E2:42_1_1_6_1_1"/><i><font color="Green" title="E55">A51</font></i>: 
<font color="Maroon" title="c22">m</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_1_1"><i><font color="Green" title="E54">A50</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">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" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">m</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6_1_1"><i><font color="Green" title="E55">A51</font></i></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">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" title="c22">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</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" title="c18">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c19">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_6_1_1_1_1"/>
<font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_1"><i><font color="Green" title="E52">A48</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" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_6_1_1_1_2"/>
<font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_6"><i><font color="Green" title="E49">A45</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_6"><i><font color="Green" title="E50">A46</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_6"><i><font color="Green" title="E51">A47</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" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">m</font>
 ;<br/><div class="add"><a NAME="E2:42_1_1_6_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_1_1"><i><font color="Green" title="E54">A50</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6_1_1"><i><font color="Green" title="E55">A51</font></i></a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_6_1_1_1_3"/>
<font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">m</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" title="E48">A44</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6_1"><i><font color="Green" title="E53">A49</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">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" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E16:42_1_1_6"/><i><font color="Green" title="E52">A52</font></i>: 
<font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <b>and </b><br/><a NAME="E17:42_1_1_6"/><i><font color="Green" title="E53">A53</font></i>: 
<font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font>
 <b>and </b><br/><a NAME="E18:42_1_1_6"/><i><font color="Green" title="E54">A54</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> holds <br/><font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E6:42_1_1_6"><i><font color="Green" title="E48">A44</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self" title="BAGORDER:def.8">Def8</a></i>;<br/><div><i><font color="Green" title="E55">A55</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" title="E56">A56</font></i>: 
not <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</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" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> or <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">i</font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_2"><i><font color="Green" title="E56">A56</font></i></a>, <a class="ref" href="ordinal1.html#T24" title="ORDINAL1:th.24">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" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</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" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1_1_6"><i><font color="Green" title="E49">A45</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1_1_6"><i><font color="Green" title="E50">A46</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1_6"><i><font color="Green" title="E51">A47</font></i></a>, <a class="txt" href="bagorder.html#E17:42_1_1_6"><i><font color="Green" title="E53">A53</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" title="E57">A57</font></i>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c22">i</font>
 ;<br/><div class="add"><b>then </b><font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E16:42_1_1_6"><i><font color="Green" title="E52">A52</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font>
<b>by </b><i><a class="txt" href="bagorder.html#E20:42_1_1"><i><font color="Green" title="E40">A18</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E16:42_1_1_6"><i><font color="Green" title="E52">A52</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_2_1_2"><i><font color="Green" title="E57">A57</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" title="E53">A53</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" title="c23">bj</font> = <font color="Maroon" title="c18">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c24">b1j</font> = <font color="Maroon" title="c19">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> as   <a href="polynom1.html#NM1" title="POLYNOM1:NM.1">bag</a> of <font color="Maroon" title="c8">j</font> <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="ref" href="bagorder.html#T21" target="_self" title="BAGORDER:th.21">Th21</a></i>;<br/><a NAME="E21:42_1_1_6"/><i><font color="Green" title="E56">A58</font></i>: 
( <font color="Maroon" title="c24">b1j</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> &amp; <font color="Maroon" title="c23">bj</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c22">i</font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:42_1_1_6"><i><font color="Green" title="E55">A55</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">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" title="c25">k</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:42_1_1_6_3"/><i><font color="Green" title="E57">A59</font></i>: 
( <font color="Maroon" title="c22">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c25">k</font> &amp; <font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font> )
 ;<br/><a NAME="E2:42_1_1_6_3"/>
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green" title="E57">A59</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><a NAME="E3:42_1_1_6_3"/><b>then </b><i><font color="Green" title="E58">A60</font></i>: 
<font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b1</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">k</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E18:42_1_1_6"><i><font color="Green" title="E54">A54</font></i></a>, <a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green" title="E57">A59</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c23">bj</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">k</font> = 
<font color="Maroon" title="c18">b</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">k</font>
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green" title="E57">A59</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c24">b1j</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c25">k</font>
<b>by </b><i><a class="txt" href="bagorder.html#E1:42_1_1_6_3"><i><font color="Green" title="E57">A59</font></i></a>, <a class="txt" href="bagorder.html#E3:42_1_1_6_3"><i><font color="Green" title="E58">A60</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">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" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1_1_6"><i><font color="Green" title="E46">A42</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1_1_6"><i><font color="Green" title="E47">A43</font></i></a>, <a class="txt" href="bagorder.html#E17:42_1_1_6"><i><font color="Green" title="E53">A53</font></i></a>, <a class="txt" href="bagorder.html#E19:42_1_1_6"><i><font color="Green" title="E55">A55</font></i></a>, <a class="txt" href="bagorder.html#E21:42_1_1_6"><i><font color="Green" title="E56">A58</font></i></a>, <a class="ref" href="bagorder.html#D8" target="_self" title="BAGORDER:def.8">Def8</a></i>;<br/></div><b>end;</b></div><a NAME="E33:42_1_1"/><b>then </b>
<font color="Maroon" title="c16">g</font> is <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#D7" title="WELLFND1:def.7">WELLFND1:def 7</a></i>;<br/><a NAME="E34:42_1_1"/><b>then </b>
not  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font></span>)</span> #) is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15" title="WELLFND1:th.15">WELLFND1:15</a></i>;<br/><a NAME="E35:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c8">j</font></span>)</span>,<span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font></span>)</span> #)
 <b>by </b><i><a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></i>;<br/><a NAME="E36:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E25:42_1_1"><i><font color="Green" title="E42">A38</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></i>;<br/><a NAME="E37:42_1_1"/><b>then </b>
not  <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c8">j</font> is <a href="wellord1.html#V2" title="WELLORD1:attr.2">well-ordering</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">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" title="E27">A2</font></i></a>, <a class="txt" href="bagorder.html#E12:42_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="bagorder.html#E15:42_1_1"><i><font color="Green" title="E37">A15</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:42_1"/><i><font color="Green" title="E31">A61</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/><a NAME="E7:42_1"/><b>then </b><i><font color="Green" title="E32">A62</font></i>: 
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/><a NAME="E8:42_1"/><i><font color="Green" title="E33">A63</font></i>: 
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:42_1"><i><font color="Green" title="E31">A61</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/><a NAME="E9:42_1"/><i><font color="Green" title="E34">A64</font></i>: 
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a>  <a href="polynom1.html#K14" title="POLYNOM1:func.14">Bags</a> <font color="Maroon" title="c1">o</font>
 <b>by </b><i><a class="txt" href="bagorder.html#E6:42_1"><i><font color="Green" title="E31">A61</font></i></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>;<br/><a NAME="E10:42_1"/><i><font color="Green" title="E35">A65</font></i>: 
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="relat_2.html#R6" title="RELAT_2:pred.6">is_connected_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:42_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="orders_1.html#T92" title="ORDERS_1:th.92">ORDERS_1:92</a></i>;<br/><a NAME="E11:42_1"/>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R1" title="WELLORD1:pred.1">is_well_founded_in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:42_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="wellord1.html#T5" title="WELLORD1:th.5">WELLORD1:5</a></i>;<br/><a NAME="E12:42_1"/><b>then </b>
 <a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font> <a href="wellord1.html#R2" title="WELLORD1:pred.2">well_orders</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="bagorder.html#K4" title="BAGORDER:func.4">InvLexOrder</a> <font color="Maroon" title="c1">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="bagorder.html#E4:42_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="bagorder.html#E7:42_1"><i><font color="Green" title="E32">A62</font></i></a>, <a class="txt" href="bagorder.html#E8:42_1"><i><font color="Green" title="E33">A63</font></i></a>, <a class="txt" href="bagorder.html#E9:42_1"><i><font color="Green" title="E34">A64</font></i></a>, <a class="txt" href="bagorder.html#E10:42_1"><i><font color="Green" title="E35">A65</font></i></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">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" title="c1">o</font>]
 <b>by </b><i><a class="ref" href="wellord1.html#T8" title="WELLORD1:th.8">WELLORD1:8</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:42"/>
for <font color="Olive" title="b1">o</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">o</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="bagorder.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>);</i><br/>


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