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<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">T1</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b3">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">s</font></span>)</span> where <font color="Olive" title="b3">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b4">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> </span>}</span>  holds <br/> ex <font color="Olive" title="b3">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b4">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b4">r</font> &amp; <font color="Olive" title="b3">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">r</font> )</span><br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> &amp; <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">s</font></span>)</span> where <font color="Olive" title="b2">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b3">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  holds <br/> ex <font color="Olive" title="b2">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b2">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">r</font> )</span><br/><b>let </b><font color="Maroon" title="c4">q</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  implies  ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> ) )</span><br/>







<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1"/>
( for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/>( not <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> ) or  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> ) )
 </span><b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="trees_a.html#T13" target="_self" title="TREES_A:th.13">Th13</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>suppose </b></a><a NAME="E1:18_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/>( not <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c5">p'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c6">r</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font><b> such that </b><br/><a NAME="E3:18_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c5">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1_1_1"/>
not <font color="Maroon" title="c5">p'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c4">q</font>
 <b>by </b><i><a class="txt" href="#E1:18_1_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E3:18_1_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:18_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:18_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:18_1_1"/>
 ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )
 <b>by </b><i><a class="txt" href="#E3:18_1_1"><i><font color="Green" title="E18">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>suppose </b></a><a NAME="E1:18_1_2"/>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">r</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Olive" title="b2">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">r</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:18_1_2"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c6">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">r</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">r</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c7">p'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c8">r'</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font><b> such that </b><br/><a NAME="E5:18_1_2"/><i><font color="Green" title="E18">A5</font></i>: 
( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c8">r'</font> &amp; <font color="Maroon" title="c7">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="#E1:18"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_2_1"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:18_1_2_1"/>
<font color="Maroon" title="c5">p</font>,<font color="Maroon" title="c7">p'</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <b>by </b><i><a class="txt" href="#E3:18_1_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E5:18_1_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="trees_a.html#T1" target="_self" title="TREES_A:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E3:18_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:18_1_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="#E5:18_1_2"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="#E1:18_1_2_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="trees_1.html#D13" title="TREES_1:def.13">TREES_1:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:18_1_2"/>
 ex <font color="Olive" title="b1">p'</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">T1</font> st <br/>( <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p'</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" title="TREES_A:func.3">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">r</font> )
 <b>by </b><i><a class="txt" href="#E3:18_1_2"><i><font color="Green" title="E17">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
