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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <span class="kw">means </span> <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><span class="lab"><font color="Green" title="E25">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">o</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">o</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">o</font>]</span><br/><span class="kw">set </span><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="E2:40_1"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D5" target="_self" title="BAGORDER:def.5">Def7</a></span>;<br/><a NAME="E3:40_1"/><span class="kw">then </span>
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T7" title="ORDERS_1:th.7">ORDERS_1:7</a></span>;<br/><a NAME="E4:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A4</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T9" title="PARTIT_2:th.9">PARTIT_2:9</a></span>;<br/><div><span class="lab"><font color="Green" title="E29">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
 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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="wellord1.html#T3" title="WELLORD1:th.3">WELLORD1:3</a></span>;<br/><span class="kw">set </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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/><span class="kw">set </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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="E3:40_1_1"/>
 not  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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> #)<span class="kw"> such that </span><br/><a NAME="E5:40_1_1"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c5">f</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#T14" title="WELLFND1:th.14">WELLFND1:14</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c7">s</font> =  <a href="pre_poly.html#K7" title="PRE_POLY:func.7">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="E7:40_1_1"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><a NAME="E8:40_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord2.html#T6" title="WELLORD2:th.6">WELLORD2:6</a></span>;<br/><a NAME="E9:40_1_1"/><span class="kw">then </span>
 <a href="yellow_1.html#K1" title="YELLOW_1:func.1">RelIncl</a> <font color="Maroon" title="c1">o</font> is  <a href="orders_1.html#V3" title="ORDERS_1:attr.3">being_linear-order</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T19" title="ORDERS_1:th.19">ORDERS_1:19</a></span>;<br/><a NAME="E10:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A9</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="orders_1.html#T37" title="ORDERS_1:th.37">ORDERS_1:37</a>, <a class="ref" href="orders_1.html#T38" title="ORDERS_1:th.38">ORDERS_1:38</a></span>;<br/><a NAME="E11:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:40_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="bagorder.html#T19" target="_self" title="BAGORDER:th.19">Th20</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/><a NAME="E4:40_1_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E5:40_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="pre_poly.html#K15" title="PRE_POLY:func.15">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> #)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E6:40_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <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="c8">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">a</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; (  for <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">k</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="bagorder.html#D6" target="_self" title="BAGORDER:def.6">Def8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:40_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="ref" href="pre_poly.html#T52" title="PRE_POLY:th.52">PRE_POLY:52</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="relat_1.html#T38" title="RELAT_1:th.38">RELAT_1:38</a></span>;<br/><a NAME="E14:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A14</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" title="FUNCT_1:th.3">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">j</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a></span>;<br/><span class="kw">defpred </span><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> ] <span class="kw">means </span> ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font> );<br/><div><span class="lab"><font color="Green" title="E37">A15</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</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>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">x</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/><span class="kw">take </span><font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c10">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">y</font>]</span><br/><span class="kw">thus </span><a NAME="E2:40_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>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E18:40_1_1"/><span class="lab"><font color="Green" title="E38">A16</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">i</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E16:40_1_1"><span class="lab"><font color="Green" title="E37">A15</font></span></a>);<br/></span><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> 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="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="nat_1.html#K6" title="NAT_1:func.6">.</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="recdef_1.html#K1" title="RECDEF_1:func.1">.</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="E19:40_1_1"/><span class="lab"><font color="Green" title="E39">A17</font></span>: 
<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="40_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">i</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="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="nat_1.html#K6" title="NAT_1:func.6">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c11">b</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K6" title="NAT_1:func.6">.</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> implies <font color="Maroon" title="c11">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_1_3"/><span class="lab"><font color="Green" title="E40">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:40_1_1_3"/><span class="lab"><font color="Green" title="E41">A19</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:40_1_1_3"/><span class="lab"><font color="Green" title="E42">A20</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">assume </span><a NAME="E4:40_1_1_3"/>
<font color="Maroon" title="c11">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:40_1_1_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1_3"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:40_1_1_3"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<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="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="kw">and </span><br/><a NAME="E8:40_1_1_3"/><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></span>;<br/>
<a NAME="E9:40_1_1_3"/><span class="lab"><font color="Green" title="E45">A23</font></span>: 
<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="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_3"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_3_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="pre_poly.html#K7" title="PRE_POLY:func.7">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#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_3"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_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="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_3"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E8:40_1_1_3"><span class="lab"><font color="Green" title="E44">A22</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_3_1_2"/>
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:40_1_1_3_1_2"/><span class="kw">then </span>
<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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:40_1_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E13:40_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="txt" href="#E7:40_1_1_3"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="pre_poly.html#D2" title="PRE_POLY:def.2">PRE_POLY:def 2</a></span>;<br/><a NAME="E3:40_1_1_3_1_2"/><span class="kw">then </span>
<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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_3"><span class="lab"><font color="Green" title="E43">A21</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E4:40_1_1_3_1_2"/><span class="kw">then </span>
<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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:40_1_1"><span class="lab"><font color="Green" title="E35">A13</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/><a NAME="E5:40_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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="pre_poly.html#K7" title="PRE_POLY:func.7">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_3"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="txt" href="#E8:40_1_1_3"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="wellord2.html#D1" title="WELLORD2:def.1">WELLORD2:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:40_1_1_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_3"><span class="lab"><font color="Green" title="E40">A18</font></span></a>, <a class="txt" href="#E8:40_1_1_3"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="ordinal1.html#T5" title="ORDINAL1:th.5">ORDINAL1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E20:40_1_1"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:40_1_1_4"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
 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="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="nat_1.html#K6" title="NAT_1:func.6">.</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="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">i</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> holds <br/><font color="Olive" title="b1">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> </span><br/><span class="kw">let </span><font color="Maroon" title="c12">b1</font> be   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font> implies <font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:40_1_1_4"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:40_1_1_4"/><span class="lab"><font color="Green" title="E43">A27</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:40_1_1_4"/><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c13">b</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">n</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/>
<a NAME="E6:40_1_1_4"/><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<font color="Maroon" title="c13">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:40_1_1_4"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E3:40_1_1_4"><span class="lab"><font color="Green" title="E43">A27</font></span></a></span>;<br/>
<a NAME="E7:40_1_1_4"/><span class="lab"><font color="Green" title="E46">A30</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:40_1_1_4"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<a NAME="E8:40_1_1_4"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E4:40_1_1_4"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">i1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E10:40_1_1_4"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E11:40_1_1_4"/><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c13">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c14">i1</font>
 <span class="kw">and </span><br/><a NAME="E12:40_1_1_4"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
 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="recdef_1.html#K1" title="RECDEF_1:func.1">.</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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:40_1_1_4"><span class="lab"><font color="Green" title="E46">A30</font></span></a>, <a class="ref" href="bagorder.html#D6" target="_self" title="BAGORDER:def.6">Def8</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_4_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><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_4_1_1"/>
<font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1_4"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E6:40_1_1_4"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E12:40_1_1_4"><span class="lab"><font color="Green" title="E49">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_4_1_2"/>
<font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:40_1_1_4"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E3:40_1_1_4"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="txt" href="#E11:40_1_1_4"><span class="lab"><font color="Green" title="E48">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_4_1_3"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> </span><br/><div class="add"><span class="kw">assume </span><a NAME="E2:40_1_1_4_1_3"/>
<font color="Maroon" title="c12">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:40_1_1_4_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_4"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E1:40_1_1_4_1_3"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1_1_4_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_4"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E10:40_1_1_4"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="txt" href="#E11:40_1_1_4"><span class="lab"><font color="Green" title="E48">A32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E21:40_1_1"/><span class="lab"><font color="Green" title="E41">A35</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E19:40_1_1"><span class="lab"><font color="Green" title="E39">A17</font></span></a>, <a class="txt" href="#E20:40_1_1"><span class="lab"><font color="Green" title="E40">A24</font></span></a>);<br/></span><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="dickson.html#D15" title="DICKSON:def.15">DICKSON:def 15</a></span>;<br/><a NAME="E23:40_1_1"/><span class="lab"><font color="Green" title="E42">A36</font></span>: 
<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="40_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="bagorder.html#D3" target="_self" title="BAGORDER:def.3">BAGORDER:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c12">n0</font> = <font color="Maroon" title="c11">n</font> 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">tn</font> = <font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n0</font>, <font color="Maroon" title="c14">tn1</font> = <font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">fn</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c12">n0</font>, <font color="Maroon" title="c16">fn1</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n0</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/>
<a NAME="E4:40_1_1_5"/><span class="lab"><font color="Green" title="E43">A37</font></span>: 
<font color="Maroon" title="c16">fn1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">fn</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<a NAME="E5:40_1_1_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">fn1</font>,<font color="Maroon" title="c15">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E7:40_1_1_5"/><span class="lab"><font color="Green" title="E44">A38</font></span>: 
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">and </span><br/><a NAME="E8:40_1_1_5"/><span class="lab"><font color="Green" title="E45">A39</font></span>: 
<font color="Maroon" title="c16">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c17">i</font>
 <span class="kw">and </span><br/><a NAME="E9:40_1_1_5"/><span class="lab"><font color="Green" title="E46">A40</font></span>: 
 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="c17">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="c16">fn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">fn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1_5"><span class="lab"><font color="Green" title="E43">A37</font></span></a>, <a class="ref" href="bagorder.html#D6" target="_self" title="BAGORDER:def.6">Def8</a></span>;<br/>
<a NAME="E10:40_1_1_5"/><span class="lab"><font color="Green" title="E47">A41</font></span>: 
 ex <font color="Olive" title="b1">bn</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c15">fn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bn</font> &amp; <font color="Maroon" title="c13">tn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bn</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/>
<a NAME="E11:40_1_1_5"/><span class="lab"><font color="Green" title="E48">A42</font></span>: 
 ex <font color="Olive" title="b1">bn1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c16">fn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bn1</font> &amp; <font color="Maroon" title="c14">tn1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bn1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_5_1_1"/>
( <font color="Maroon" title="c17">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="c17">i</font> or <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_5_1_1_1"/>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_5_1_1_1"/>
<font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:40_1_1_5"><span class="lab"><font color="Green" title="E45">A39</font></span></a>, <a class="txt" href="#E10:40_1_1_5"><span class="lab"><font color="Green" title="E47">A41</font></span></a>, <a class="txt" href="#E11:40_1_1_5"><span class="lab"><font color="Green" title="E48">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_5_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="c17">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_5_1_1_2"/>
<font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E7:40_1_1_5"><span class="lab"><font color="Green" title="E44">A38</font></span></a>, <a class="txt" href="#E8:40_1_1_5"><span class="lab"><font color="Green" title="E45">A39</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_5_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_5_1_1_3"/>
<font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_5_1_1_3"/>
<font color="Maroon" title="c14">tn1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">tn</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E9:40_1_1_5"><span class="lab"><font color="Green" title="E46">A40</font></span></a>, <a class="txt" href="#E10:40_1_1_5"><span class="lab"><font color="Green" title="E47">A41</font></span></a>, <a class="txt" href="#E11:40_1_1_5"><span class="lab"><font color="Green" title="E48">A42</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:40_1_1_5"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">set </span><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c8">j</font>;<br/><span class="kw">set </span><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/><span class="kw">set </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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="E24:40_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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bagorder.html#D5" target="_self" title="BAGORDER:def.5">Def7</a></span>;<br/><a NAME="E25:40_1_1"/><span class="kw">then </span>
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T7" title="ORDERS_1:th.7">ORDERS_1:7</a></span>;<br/><a NAME="E26:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A43</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partit_2.html#T9" title="PARTIT_2:th.9">PARTIT_2:9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E28:40_1_1"/><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="nat_1.html#K6" title="NAT_1:func.6">.</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:40_1_1"><span class="lab"><font color="Green" title="E42">A36</font></span></a>, <a class="ref" href="bagorder.html#T10" target="_self" title="BAGORDER:th.10">Th11</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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><span class="lab"><font color="Green" title="E45">A45</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c15">x</font></span>)</span> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> ;<br/><span class="kw">reconsider </span><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="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c8">j</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="bagorder.html#T20" target="_self" title="BAGORDER:th.20">Th21</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">y</font> = <font color="Maroon" title="c17">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D12" title="PRE_POLY:def.12">PRE_POLY:def 12</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c18">y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c19">y</font>]</span><br/><span class="kw">thus </span><a NAME="E4:40_1_1_6"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c15">x</font>,<font color="Maroon" title="c19">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">consider </span><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>,<span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c8">j</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E31:40_1_1"/><span class="lab"><font color="Green" title="E46">A46</font></span>: 
 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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E29:40_1_1"><span class="lab"><font color="Green" title="E45">A45</font></span></a>);<br/></span><span class="kw">reconsider </span><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="pre_poly.html#K15" title="PRE_POLY:func.15">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="40_1_1_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">k</font> &amp; <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">k0</font> = <font color="Maroon" title="c17">k</font> 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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c19">b</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font><span class="kw"> such that </span><br/><a NAME="E3:40_1_1_7"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:40_1_1_7"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c18">k0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:40_1_1"><span class="lab"><font color="Green" title="E46">A46</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c20">b1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font><span class="kw"> such that </span><br/><a NAME="E6:40_1_1_7"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
<font color="Maroon" title="c20">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:40_1_1_7"/><span class="lab"><font color="Green" title="E50">A50</font></span>: 
<font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:40_1_1"><span class="lab"><font color="Green" title="E46">A46</font></span></a></span>;<br/><a NAME="E8:40_1_1_7"/>
<font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><a NAME="E9:40_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A51</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c20">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E10:40_1_1_7"/><span class="lab"><font color="Green" title="E52">A52</font></span>: 
 ex <font color="Olive" title="b1">bb</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bb</font> &amp; <font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c18">k0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bb</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/><a NAME="E11:40_1_1_7"/><span class="lab"><font color="Green" title="E53">A53</font></span>: 
 ex <font color="Olive" title="b1">bb1</font> being   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c1">o</font> st <br/>( <font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bb1</font> &amp; <font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">bb1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c8">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1"><span class="lab"><font color="Green" title="E38">A16</font></span></a></span>;<br/><a NAME="E12:40_1_1_7"/><span class="lab"><font color="Green" title="E54">A54</font></span>: 
<font color="Maroon" title="c10">t</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c14">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_1_1"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="kw">thus </span><a NAME="E13:40_1_1_7"/>
<font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">k</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:40_1_1_7_1"/><span class="lab"><font color="Green" title="E55">A55</font></span>: 
 not <font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c17">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:40_1_1_7_1"/><span class="lab"><font color="Green" title="E56">A56</font></span>: 
<font color="Maroon" title="c1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c19">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E3:40_1_1_7_1"/><span class="lab"><font color="Green" title="E57">A57</font></span>: 
<font color="Maroon" title="c1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c20">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_7_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c21">m</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font> implies <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">assume </span><a NAME="E1:40_1_1_7_1_1"/><span class="lab"><font color="Green" title="E58">A58</font></span>: 
<font color="Maroon" title="c21">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_7_1_1_1"/>
( <font color="Maroon" title="c21">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="c21">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="c21">m</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_7_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A59</font></span>: 
<font color="Maroon" title="c21">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:40_1_1_7_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">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="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_7_1_1_1_1"/>
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E7:40_1_1_7"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E1:40_1_1_7_1"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="txt" href="#E1:40_1_1_7_1_1_1_1"><span class="lab"><font color="Green" title="E59">A59</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_7_1_1_1_2"/>
<font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_7_1_1_1_2"/>
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_1_1"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E10:40_1_1_7"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="txt" href="#E11:40_1_1_7"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="txt" href="#E12:40_1_1_7"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_7_1_1_1_3"/><span class="lab"><font color="Green" title="E59">A60</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><br/><div class="add"><a NAME="E2:40_1_1_7_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E1:40_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A58</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_7_1_1_1_3"/>
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E1:40_1_1_7_1_1"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="txt" href="#E1:40_1_1_7_1_1_1_3"><span class="lab"><font color="Green" title="E59">A60</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:40_1_1_7_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:40_1_1_7"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E2:40_1_1_7_1"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="txt" href="#E3:40_1_1_7_1"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:40_1_1_7"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">i</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><span class="kw"> such that </span><br/><a NAME="E16:40_1_1_7"/><span class="lab"><font color="Green" title="E55">A61</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">and </span><br/><a NAME="E17:40_1_1_7"/><span class="lab"><font color="Green" title="E56">A62</font></span>: 
<font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">and </span><br/><a NAME="E18:40_1_1_7"/><span class="lab"><font color="Green" title="E57">A63</font></span>: 
 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="c21">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="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E9:40_1_1_7"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="ref" href="bagorder.html#D6" target="_self" title="BAGORDER:def.6">Def8</a></span>;<br/><div><span class="lab"><font color="Green" title="E58">A64</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_7_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1_1_7_2"/><span class="lab"><font color="Green" title="E59">A65</font></span>: 
 not <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1_1_7_2_1"/>
( <font color="Maroon" title="c21">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="c21">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1_7_2"><span class="lab"><font color="Green" title="E59">A65</font></span></a>, <a class="ref" href="ordinal1.html#T14" title="ORDINAL1:th.14">ORDINAL1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_7_2_1_1"/>
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:40_1_1_7_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:40_1_1"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E10:40_1_1_7"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="txt" href="#E11:40_1_1_7"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="txt" href="#E12:40_1_1_7"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="txt" href="#E17:40_1_1_7"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1_7_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_1_7_2_1_2"/><span class="lab"><font color="Green" title="E60">A66</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> = 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E3:40_1_1_7"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E16:40_1_1_7"><span class="lab"><font color="Green" title="E55">A61</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:40_1_1"><span class="lab"><font color="Green" title="E41">A35</font></span></a>, <a class="txt" href="#E6:40_1_1_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E16:40_1_1_7"><span class="lab"><font color="Green" title="E55">A61</font></span></a>, <a class="txt" href="#E1:40_1_1_7_2_1_2"><span class="lab"><font color="Green" title="E60">A66</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E3:40_1_1_7_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:40_1_1_7"><span class="lab"><font color="Green" title="E56">A62</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c22">bj</font> = <font color="Maroon" title="c19">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="c23">b1j</font> = <font color="Maroon" title="c20">b1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c8">j</font> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of <font color="Maroon" title="c8">j</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="ref" href="bagorder.html#T20" target="_self" title="BAGORDER:th.20">Th21</a></span>;<br/><a NAME="E21:40_1_1_7"/><span class="lab"><font color="Green" title="E59">A67</font></span>: 
<font color="Maroon" title="c23">b1j</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1_1_7"><span class="lab"><font color="Green" title="E58">A64</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><a NAME="E22:40_1_1_7"/><span class="lab"><font color="Green" title="E60">A68</font></span>: 
<font color="Maroon" title="c22">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c21">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:40_1_1_7"><span class="lab"><font color="Green" title="E58">A64</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_7_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c24">k</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">k</font> &amp; <font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font> implies <font color="Maroon" title="c22">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b1j</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:40_1_1_7_3"/><span class="lab"><font color="Green" title="E61">A69</font></span>: 
<font color="Maroon" title="c21">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">and </span><br/><a NAME="E2:40_1_1_7_3"/><span class="lab"><font color="Green" title="E62">A70</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c22">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b1j</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font></span><br/><a NAME="E3:40_1_1_7_3"/>
<font color="Maroon" title="c24">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a>, <a class="txt" href="#E2:40_1_1_7_3"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></span>;<br/><a NAME="E4:40_1_1_7_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A71</font></span>: 
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:40_1_1_7"><span class="lab"><font color="Green" title="E57">A63</font></span></a>, <a class="txt" href="#E1:40_1_1_7_3"><span class="lab"><font color="Green" title="E61">A69</font></span></a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c22">bj</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font> = 
<font color="Maroon" title="c19">b</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_7_3"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c23">b1j</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1_1_7_3"><span class="lab"><font color="Green" title="E62">A70</font></span></a>, <a class="txt" href="#E4:40_1_1_7_3"><span class="lab"><font color="Green" title="E63">A71</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E24:40_1_1_7"/>
<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="nat_1.html#K6" title="NAT_1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">g</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_1_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E7:40_1_1_7"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="txt" href="#E17:40_1_1_7"><span class="lab"><font color="Green" title="E56">A62</font></span></a>, <a class="txt" href="#E19:40_1_1_7"><span class="lab"><font color="Green" title="E58">A64</font></span></a>, <a class="txt" href="#E21:40_1_1_7"><span class="lab"><font color="Green" title="E59">A67</font></span></a>, <a class="txt" href="#E22:40_1_1_7"><span class="lab"><font color="Green" title="E60">A68</font></span></a>, <a class="ref" href="bagorder.html#D6" target="_self" title="BAGORDER:def.6">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E34:40_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">g</font> is  <a href="wellfnd1.html#V3" title="WELLFND1:attr.3">descending</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D6" title="WELLFND1:def.6">WELLFND1:def 6</a></span>;<br/><a NAME="E35:40_1_1"/><span class="kw">then </span>
 not  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <span class="p1">(<span class="default"><a href="pre_poly.html#K15" title="PRE_POLY:func.15">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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#T14" title="WELLFND1:th.14">WELLFND1:14</a></span>;<br/><a NAME="E36:40_1_1"/><span class="kw">then </span>
 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="pre_poly.html#K15" title="PRE_POLY:func.15">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> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellfnd1.html#D2" title="WELLFND1:def.2">WELLFND1:def 2</a></span>;<br/><a NAME="E37:40_1_1"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:40_1_1"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></span>;<br/><a NAME="E38:40_1_1"/><span class="kw">then </span>
 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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T4" title="WELLORD1:th.4">WELLORD1:4</a></span>;<br/><span class="kw">hence </span><a NAME="E39:40_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E11:40_1_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E14:40_1_1"><span class="lab"><font color="Green" title="E36">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:40_1"/><span class="lab"><font color="Green" title="E30">A72</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_1.html#T12" title="ORDERS_1:th.12">ORDERS_1:12</a></span>;<br/><a NAME="E7:40_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A73</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></span>;<br/><a NAME="E8:40_1"/><span class="lab"><font color="Green" title="E32">A74</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E30">A72</font></span></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></span>;<br/><a NAME="E9:40_1"/><span class="lab"><font color="Green" title="E33">A75</font></span>: 
 <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="pre_poly.html#K15" title="PRE_POLY:func.15">Bags</a> <font color="Maroon" title="c1">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1"><span class="lab"><font color="Green" title="E30">A72</font></span></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></span>;<br/><a NAME="E10:40_1"/><span class="lab"><font color="Green" title="E34">A76</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="orders_1.html#T7" title="ORDERS_1:th.7">ORDERS_1:7</a></span>;<br/><a NAME="E11:40_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:40_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="wellord1.html#T3" title="WELLORD1:th.3">WELLORD1:3</a></span>;<br/><a NAME="E12:40_1"/><span class="kw">then </span>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:40_1"><span class="lab"><font color="Green" title="E31">A73</font></span></a>, <a class="txt" href="#E8:40_1"><span class="lab"><font color="Green" title="E32">A74</font></span></a>, <a class="txt" href="#E9:40_1"><span class="lab"><font color="Green" title="E33">A75</font></span></a>, <a class="txt" href="#E10:40_1"><span class="lab"><font color="Green" title="E34">A76</font></span></a>, <a class="ref" href="wellord1.html#D5" title="WELLORD1:def.5">WELLORD1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E13:40_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">o</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="wellord1.html#T4" title="WELLORD1:th.4">WELLORD1:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E2:40"/>
 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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E25">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
