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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</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">c</font>, <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b2">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> holds <br/> ex <font color="Olive" title="b5">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b5">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">c</font>,<font color="Olive" title="b5">e</font>,<font color="Olive" title="b2">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <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="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> implies  ex <font color="Olive" title="b1">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E6">A1</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
( (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> ) )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">JORDAN17:def 1</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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_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="E1:18_1"/>
( (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> ) or (  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:18_1_1"/><span class="lab"><font color="Green" title="E8">A3</font></span>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> )
 <span class="kw">and </span><br/><a NAME="E2:18_1_1"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:18_1_1"/><span class="lab"><font color="Green" title="E10">A5</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E2:18_1_1"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="jordan17.html#T8" target="_self" title="JORDAN17:th.8">Th8</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">e</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E5:18_1_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E8">A3</font></span></a>, <a class="ref" href="jordan6.html#T58" title="JORDAN6:th.58">JORDAN6:58</a></span>;<br/><span class="kw">hence </span><a NAME="E6:18_1_1"/>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_1"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">Def1</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="18_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
(  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> )
 <span class="kw">and </span><br/><a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:18_1_2"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="jordan17.html#T8" target="_self" title="JORDAN17:th.8">Th8</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">e</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">thus </span><a NAME="E5:18_1_2"/>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E4:18_1_2"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">Def1</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="18_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:18_1_3"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E2:18_1_3"/><span class="lab"><font color="Green" title="E9">A10</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E3:18_1_3"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:18_1_3"/><span class="lab"><font color="Green" title="E10">A11</font></span>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font> &amp;  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:18_1_3"><span class="lab"><font color="Green" title="E8">A9</font></span></a>, <a class="ref" href="jordan17.html#T8" target="_self" title="JORDAN17:th.8">Th8</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">e</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">thus </span><a NAME="E6:18_1_3"/>
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_3"><span class="lab"><font color="Green" title="E9">A10</font></span></a>, <a class="txt" href="#E5:18_1_3"><span class="lab"><font color="Green" title="E10">A11</font></span></a>, <a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">Def1</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="18_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:18_1_4"/><span class="lab"><font color="Green" title="E8">A12</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E2:18_1_4"/><span class="lab"><font color="Green" title="E9">A13</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">and </span><br/><a NAME="E3:18_1_4"/><span class="lab"><font color="Green" title="E10">A14</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E4:18_1_4"/>
 ex <font color="Olive" title="b1">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_4_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:18_1_4_1"/><span class="lab"><font color="Green" title="E11">A15</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_4"><span class="lab"><font color="Green" title="E9">A13</font></span></a>, <a class="txt" href="#E3:18_1_4"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="ref" href="jordan6.html#T58" title="JORDAN6:th.58">JORDAN6:58</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_4_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="E1:18_1_4_1_1"/>
( <font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font> or <font color="Maroon" title="c3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:18_1_4_1_1_1"/><span class="lab"><font color="Green" title="E12">A16</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><a NAME="E2:18_1_4_1_1_1"/>
<font color="Maroon" title="c2">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_4"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><span class="kw"> such that </span><br/><a NAME="E4:18_1_4_1_1_1"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">and </span><br/><a NAME="E5:18_1_4_1_1_1"/><span class="lab"><font color="Green" title="E14">A18</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan17.html#T7" target="_self" title="JORDAN17:th.7">Th7</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">e</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">thus </span><a NAME="E6:18_1_4_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_4_1_1_1"><span class="lab"><font color="Green" title="E13">A17</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"> ( <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">thus </span><a NAME="E7:18_1_4_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E1:18_1_4_1_1_1"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E5:18_1_4_1_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="jordan7.html#T2" title="JORDAN7:th.2">JORDAN7:2</a></span>; <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="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font></span><br/><span class="kw">thus </span><a NAME="E8:18_1_4_1_1_1"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_4"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="txt" href="#E1:18_1_4_1"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E5:18_1_4_1_1_1"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">Def1</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="18_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:18_1_4_1_1_2"/><span class="lab"><font color="Green" title="E12">A19</font></span>: 
<font color="Maroon" title="c3">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</font>
 ; <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">e</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b1">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c6">e</font> =  <a href="pscomp_1.html#K18" title="PSCOMP_1:func.18">W-min</a> <font color="Maroon" title="c1">P</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><a NAME="E2:18_1_4_1_1_2"/>
<font color="Maroon" title="c3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_4"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="ref" href="jordan7.html#T5" title="JORDAN7:th.5">JORDAN7:5</a></span>;<br/><a NAME="E3:18_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A20</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan7.html#T3" title="JORDAN7:th.3">JORDAN7:3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_4_1_1_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:18_1_4_1_1_2_1"/>
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_4"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="txt" href="#E2:18_1_4"><span class="lab"><font color="Green" title="E9">A13</font></span></a>, <a class="ref" href="jordan6.html#T58" title="JORDAN6:th.58">JORDAN6:58</a></span>;<br/><a NAME="E2:18_1_4_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A21</font></span>: 
 <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_4"><span class="lab"><font color="Green" title="E10">A14</font></span></a>, <a class="ref" href="jordan6.html#T58" title="JORDAN6:th.58">JORDAN6:58</a></span>;<br/><span class="kw">assume </span><a NAME="E3:18_1_4_1_1_2_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</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/><span class="kw">hence </span><a NAME="E4:18_1_4_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="txt" href="#E3:18_1_4_1_1_2"><span class="lab"><font color="Green" title="E13">A20</font></span></a>, <a class="txt" href="#E2:18_1_4_1_1_2_1"><span class="lab"><font color="Green" title="E14">A21</font></span></a>, <a class="ref" href="jordan6.html#T57" title="JORDAN6:th.57">JORDAN6:57</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="E5:18_1_4_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">c</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="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font> )</span><br/><span class="kw">thus </span><a NAME="E6:18_1_4_1_1_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_4_1_1_2"><span class="lab"><font color="Green" title="E12">A19</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"> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font></span><br/><span class="kw">thus </span><a NAME="E7:18_1_4_1_1_2"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c</font>,<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">d</font> <a href="jordan17.html#R1" title="JORDAN17:pred.1">are_in_this_order_on</a> <font color="Maroon" title="c1">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_4"><span class="lab"><font color="Green" title="E8">A12</font></span></a>, <a class="txt" href="#E1:18_1_4_1"><span class="lab"><font color="Green" title="E11">A15</font></span></a>, <a class="txt" href="#E3:18_1_4_1_1_2"><span class="lab"><font color="Green" title="E13">A20</font></span></a>, <a class="ref" href="jordan17.html#D1" target="_self" title="JORDAN17:def.1">Def1</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
