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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_005.html#V1" title="GLIB_005:attr.1">natural-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</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">EL</font> being  <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">source</font>, <font color="Olive" title="b3">sink</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b2">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">sink</font> &amp; <font color="Olive" title="b1">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font> &amp; <font color="Olive" title="b4">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font> &amp; <font color="Olive" title="b4">P</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Olive" title="b1">EL</font> holds <br/> <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Olive" title="b1">EL</font>,<font color="Olive" title="b4">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font></span><br/><b>let </b><font color="Maroon" title="c2">EL</font> be   <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</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">source</font>, <font color="Olive" title="b2">sink</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">P</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">sink</font> &amp; <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> &amp; <font color="Olive" title="b3">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> &amp; <font color="Olive" title="b3">P</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> holds <br/> <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Olive" title="b3">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font></span><br/><b>let </b><font color="Maroon" title="c3">source</font>, <font color="Maroon" title="c4">sink</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>  st <font color="Maroon" title="c3">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Olive" title="b1">P</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> holds <br/> <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Olive" title="b1">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span><br/><b>let </b><font color="Maroon" title="c5">P</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c5">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c5">P</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font> implies  <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c3">source</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green" title="E39">A1a</font></i>: 
<font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green" title="E40">A1b</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green" title="E41">A1c</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="glib_005.html#R5" title="GLIB_005:pred.5">is_augmenting_wrt</a> <font color="Maroon" title="c2">EL</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span><br/>

<b>set </b><font color="Maroon" title="c6">EL2</font> =  <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:45_1"/>
( <font color="Maroon" title="c3">source</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c4">sink</font> is   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A1a</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</a></i>; <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">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> ) ) &amp; ( for <font color="Olive" title="b1">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> ) )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> implies (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> ) )</span><br/><b>assume </b><a NAME="E1:45_1_1"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )</span><br/><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green" title="E43">A5</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )
 <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A1a</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1_1"/>
(  not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  or <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_1"/>
not <font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_1_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E1:45_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="#E2:45_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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="45_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_1_1_1_2"/><i><font color="Green" title="E44">A6</font></i>: 
( <font color="Maroon" title="c8">n</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> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T101" title="GLIB_001:th.101">GLIB_001:101</a></i>;<br/><a NAME="E4:45_1_1_1_1_2"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E5:45_1_1_1_1_2"/>
( <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E40">A1b</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/><a NAME="E6:45_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E46">A8</font></i>: 
not <font color="Maroon" title="c5">P</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/><div><i><font color="Green" title="E47">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_1_1_1_2_1"/>
( <font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c1">G</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="E2:45_1_1_1_1_2_1"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:45_1_1_1_1_2_1"/><i><font color="Green" title="E49">A12</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:45_1_1_1_1_2_1"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></i>;<br/><a NAME="E5:45_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="#E2:45_1_1_1_1_2_1"><i><font color="Green" title="E48">A11</font></i></a>, <a class="txt" href="#E3:45_1_1_1_1_2_1"><i><font color="Green" title="E49">A12</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><a NAME="E6:45_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E40">A1b</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><b>hence </b><a NAME="E7:45_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="#E2:45_1_1_1_1_2_1"><i><font color="Green" title="E48">A11</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><b>set </b><font color="Maroon" title="c9">PFS</font> = <font color="Maroon" title="c5">P</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font>;<br/><b>set </b><font color="Maroon" title="c10">n1div2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2;<br/><a NAME="E8:45_1_1_1_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:45_1_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_001.html#K12" title="GLIB_001:func.12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="glib_001.html#T78" title="GLIB_001:th.78">GLIB_001:78</a></i>;<br/><a NAME="E10:45_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def18</a></i>;<br/><a NAME="E11:45_1_1_1_1_2"/>
2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T22" title="PEPIN:th.22">PEPIN:22</a></i>;<br/><a NAME="E12:45_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/><a NAME="E13:45_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E50">A15</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ;<br/><a NAME="E14:45_1_1_1_1_2"/><i><font color="Green" title="E51">A16</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="txt" href="#E12:45_1_1_1_1_2"><i><font color="Green" title="E49">A14</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> or <font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c1">G</font> )
 </span><b>by </b><i><a class="txt" href="#E4:45_1_1_1_1_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_2_2_1_1"/><i><font color="Green" title="E52">A17</font></i>: 
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )</span><br/><div class="add"><a NAME="E2:45_1_1_1_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E53">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><b>thus </b><a NAME="E3:45_1_1_1_1_2_2_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span><br/><a NAME="E4:45_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="#E10:45_1_1_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="txt" href="#E13:45_1_1_1_1_2"><i><font color="Green" title="E50">A15</font></i></a>, <a class="txt" href="#E14:45_1_1_1_1_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="txt" href="#E1:45_1_1_1_1_2_2_1_1"><i><font color="Green" title="E52">A17</font></i></a>, <a class="ref" href="glib_005.html#D15" target="_self" title="GLIB_005:def.15">Def18</a></i>;<br/><a NAME="E5:45_1_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:45_1_1_1_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:45_1_1_1_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E6:45_1_1_1_1_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="glib_005.html#D16" target="_self" title="GLIB_005:def.16">Def19</a></i>;<br/><a NAME="E7:45_1_1_1_1_2_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:45_1_1_1_1_2_2_1_1"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E8:45_1_1_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 ; <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="45_1_1_1_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c7">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>,<font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )</span><br/><div class="add"><a NAME="E2:45_1_1_1_1_2_2_1_2"/><b>then </b><i><font color="Green" title="E52">A20</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E3:45_1_1_1_1_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="#E7:45_1_1_1_1_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><b>thus </b><a NAME="E3:45_1_1_1_1_2_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span><br/><a NAME="E4:45_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E2:45_1_1_1_1_2_2_1_2"><i><font color="Green" title="E52">A20</font></i></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/><b>hence </b><a NAME="E5:45_1_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_1"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</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><b>end;</b></div><b>hence </b><a NAME="E16:45_1_1_1_1_2"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )
 ; <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><b>end;</b></div><b>hence </b><a NAME="E4:45_1_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c7">e</font> )
 ; <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="E3:45_1"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> holds <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">e</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">v</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Olive" title="b1">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> )</span><br/><b>assume </b><a NAME="E4:45_1"/><i><font color="Green" title="E42">A21</font></i>: 
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">source</font> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">sink</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><a NAME="E5:45_1"/><i><font color="Green" title="E43">A23</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A1a</font></i></a>, <a class="txt" href="#E4:45_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  or  not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c8">n</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_1"/><i><font color="Green" title="E44">A24</font></i>: 
( <font color="Maroon" title="c8">n</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> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></i>;<br/><div><i><font color="Green" title="E45">A25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_1"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then </b><font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c5">P</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="glib_001.html#D7" title="GLIB_001:def.7">GLIB_001:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c4">sink</font>
<b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E40">A1b</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:45_1"><i><font color="Green" title="E42">A21</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><a NAME="E5:45_1_2_1_1"/><b>then </b><i><font color="Green" title="E46">A26</font></i>: 
<font color="Maroon" title="c8">n</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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E6:45_1_2_1_1"/><i><font color="Green" title="E47">A27</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><div><i><font color="Green" title="E48">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_2"/>
<font color="Maroon" title="c8">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then </b><font color="Maroon" title="c7">v</font> = 
<font color="Maroon" title="c5">P</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="glib_001.html#D6" title="GLIB_001:def.6">GLIB_001:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">source</font>
<b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E40">A1b</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:45_1"><i><font color="Green" title="E42">A21</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><a NAME="E8:45_1_2_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="#E6:45_1_2_1_1"><i><font color="Green" title="E47">A27</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E9:45_1_2_1_1"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">n2</font> = <font color="Maroon" title="c8">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> as  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><b>set </b><font color="Maroon" title="c10">e1</font> = <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c11">e2</font> = <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c12">T</font> = <font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font>;<br/><a NAME="E11:45_1_2_1_1"/><i><font color="Green" title="E49">A29</font></i>: 
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E12:45_1_2_1_1"/><b>then </b><i><font color="Green" title="E50">A30</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E13:45_1_2_1_1"/><i><font color="Green" title="E51">A31</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></i>;<br/><a NAME="E14:45_1_2_1_1"/><i><font color="Green" title="E52">A32</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a></i>;<br/><div><i><font color="Green" title="E53">A33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_3"/><i><font color="Green" title="E54">A34</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</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="E2:45_1_2_1_1_3"/>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_3"><i><font color="Green" title="E54">A34</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><i><font color="Green" title="E54">A35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_4"/><i><font color="Green" title="E55">A36</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:45_1_2_1_1_4"/><i><font color="Green" title="E56">A37</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E3:45_1_2_1_1_4"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_4"><i><font color="Green" title="E55">A36</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_4"><i><font color="Green" title="E56">A37</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><i><font color="Green" title="E55">A38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_5"/><i><font color="Green" title="E56">A39</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then </b><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_5"><i><font color="Green" title="E56">A39</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>
;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E15:45_1_2_1_1"><i><font color="Green" title="E53">A33</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><i><font color="Green" title="E56">A40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_6"/><i><font color="Green" title="E57">A41</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then </b><i><font color="Green" title="E58">A42</font></i>: <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>
<br/>.= 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_6"><i><font color="Green" title="E57">A41</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>
;<br/><a NAME="E3:45_1_2_1_1_6"/><i><font color="Green" title="E59">A43</font></i>: 
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E4:45_1_2_1_1_6"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_6"><i><font color="Green" title="E58">A42</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_6"><i><font color="Green" title="E59">A43</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><a NAME="E19:45_1_2_1_1"/>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E20:45_1_2_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <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> <font color="Maroon" title="c5">P</font> )
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E21:45_1_2_1_1"/><b>then </b><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/><div><i><font color="Green" title="E58">A45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  or <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  ) &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> implies  not <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_7"/><i><font color="Green" title="E59">A46</font></i>: 
( ( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  or <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  ) &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> </span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_7_1"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c14">v1</font>, <font color="Maroon" title="c15">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c16">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_1_7_1"/><i><font color="Green" title="E60">A47</font></i>: 
( <font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m</font> &amp; <font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c13">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">v1</font>,<font color="Maroon" title="c15">v2</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/><a NAME="E4:45_1_2_1_1_7_1"/><i><font color="Green" title="E61">A48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><div><i><font color="Green" title="E62">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_7_1_1_1"/>
( <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  or <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 </span><b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_7"><i><font color="Green" title="E59">A46</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_1_1_1"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> or <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/><div class="add"><a NAME="E2:45_1_2_1_1_7_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T59" title="GLIB_000:th.59">GLIB_000:59</a></i>;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_7_1_1_1_1"/>
( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> or <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</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="45_1_2_1_1_7_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_1_1_2"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> or <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )</span><br/><div class="add"><a NAME="E2:45_1_2_1_1_7_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T61" title="GLIB_000:th.61">GLIB_000:61</a></i>;<br/><b>hence </b><a NAME="E3:45_1_2_1_1_7_1_1_1_2"/>
( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> or <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</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><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_7_1_2_1"/>
( <font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> or <font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 </span><b>by </b><i><a class="txt" href="#E5:45_1_2_1_1_7_1"><i><font color="Green" title="E62">A49</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_1"/><i><font color="Green" title="E63">A50</font></i>: 
<font color="Maroon" title="c14">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_7_1_2_1_1_1_1"/>
( <font color="Maroon" title="c16">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c16">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_7_1_2_1_1"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_7_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c16">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_7"><i><font color="Green" title="E59">A46</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</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="45_1_2_1_1_7_1_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_1_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1_7_1"><i><font color="Green" title="E61">A48</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_7_1_2_1_1"><i><font color="Green" title="E63">A50</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_7_1_2_1_1"/>
contradiction
 ; <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="45_1_2_1_1_7_1_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_2"/><i><font color="Green" title="E63">A51</font></i>: 
<font color="Maroon" title="c15">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_7_1_2_1_2_1_1"/>
( <font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_7_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_2_1_1_1"/>
<font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_7_1_2_1_2"><i><font color="Green" title="E63">A51</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_7_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_2_1_1_2"/>
<font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_7"><i><font color="Green" title="E59">A46</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</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="45_1_2_1_1_7_1_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_7_1_2_1_2_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_7_1_2_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_7_1"><i><font color="Green" title="E60">A47</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_7_1_2_1_2"><i><font color="Green" title="E63">A51</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_7_1_2_1_2"/>
contradiction
 ; <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><b>end;</b></div><b>hence </b><a NAME="E7:45_1_2_1_1_7_1"/>
contradiction
 ; <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="E3:45_1_2_1_1_7"/>
not <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <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="hs2(this)" href="javascript:()" title="45_1_2_1_1_8"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1"/>
( ( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> ) or ( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> ) or ( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> ) or ( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> ) )
 </span><b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E12:45_1_2_1_1"><i><font color="Green" title="E50">A30</font></i></a>, <a class="txt" href="#E13:45_1_2_1_1"><i><font color="Green" title="E51">A31</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E59">A52</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><b>set </b><font color="Maroon" title="c13">XIN</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">XOUT</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E60">A53</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_1"><i><font color="Green" title="E59">A52</font></i></a>, <a class="ref" href="glib_000.html#T60" title="GLIB_000:th.60">GLIB_000:60</a>, <a class="ref" href="glib_000.html#T62" title="GLIB_000:th.62">GLIB_000:62</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E61">A54</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then </b><i><font color="Green" title="E62">A55</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c15">XIN</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E6:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E63">A56</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then </b><i><font color="Green" title="E64">A57</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c16">XOUT</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E65">A58</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_1_3"/>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_1_3"/><b>then </b><i><font color="Green" title="E66">A59</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_1_3"/><b>then </b><i><font color="Green" title="E67">A60</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T59" title="GLIB_000:th.59">GLIB_000:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_1"/><i><font color="Green" title="E68">A61</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1"><i><font color="Green" title="E61">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_1"><i><font color="Green" title="E68">A61</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E11:45_1_2_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="#E14:45_1_2_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_1"><i><font color="Green" title="E68">A61</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>
;<br/> <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="45_1_2_1_1_8_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2"/><i><font color="Green" title="E68">A62</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1"><i><font color="Green" title="E61">A54</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E69">A63</font></i>: <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1_3"><i><font color="Green" title="E66">A59</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c18">v1</font>, <font color="Maroon" title="c19">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c20">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_1_8_1_1_3_1_1_2_2"/><i><font color="Green" title="E70">A64</font></i>: 
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c19">v2</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/><a NAME="E4:45_1_2_1_1_8_1_1_3_1_1_2_2"/><i><font color="Green" title="E71">A65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v1</font> or <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font> )
 </span><b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1_3"><i><font color="Green" title="E67">A60</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"/><i><font color="Green" title="E72">A66</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"><i><font color="Green" title="E72">A66</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E73">A67</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3"><i><font color="Green" title="E67">A60</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/><i><font color="Green" title="E74">A68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font>
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="ref" href="glib_001.html#T1" title="GLIB_001:th.1">GLIB_001:1</a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"><i><font color="Green" title="E73">A67</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_2"><i><font color="Green" title="E74">A68</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E71">A65</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"><i><font color="Green" title="E72">A66</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_1"/>
contradiction
 ; <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="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"/><i><font color="Green" title="E72">A69</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"><i><font color="Green" title="E72">A69</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_2"><i><font color="Green" title="E68">A62</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</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="45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2_2"><i><font color="Green" title="E70">A64</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"><i><font color="Green" title="E72">A69</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_8_1_1_3_1_1_2_2_1_1_2"/>
contradiction
 ; <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><b>end;</b></div><b>hence </b><a NAME="E6:45_1_2_1_1_8_1_1_3_1_1_2_2"/>
contradiction
 ; <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="E5:45_1_2_1_1_8_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_1_3"><i><font color="Green" title="E66">A59</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_3_1_1_2"><i><font color="Green" title="E69">A63</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1_3"><i><font color="Green" title="E66">A59</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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><a NAME="E11:45_1_2_1_1_8_1_1"/><b>then </b><i><font color="Green" title="E66">A70</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XIN</font>
 <b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_1"><i><font color="Green" title="E62">A55</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_1"><i><font color="Green" title="E65">A58</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><a NAME="E12:45_1_2_1_1_8_1_1"/><i><font color="Green" title="E67">A71</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green" title="E68">A72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_1_4"/>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_1_4"/><b>then </b><i><font color="Green" title="E69">A73</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_1_4"/><b>then </b><i><font color="Green" title="E70">A74</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T61" title="GLIB_000:th.61">GLIB_000:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_1"/><i><font color="Green" title="E71">A75</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:45_1_2_1_1_8_1_1"><i><font color="Green" title="E63">A56</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_1"><i><font color="Green" title="E71">A75</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_1"><i><font color="Green" title="E71">A75</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>
;<br/> <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="45_1_2_1_1_8_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2"/><i><font color="Green" title="E71">A76</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:45_1_2_1_1_8_1_1"><i><font color="Green" title="E63">A56</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E72">A77</font></i>: <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c18">v1</font>, <font color="Maroon" title="c19">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c20">m</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_1_8_1_1_4_1_1_2_2"/><i><font color="Green" title="E73">A78</font></i>: 
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <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> <font color="Maroon" title="c5">P</font> &amp; <font color="Maroon" title="c18">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m</font> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c19">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c18">v1</font>,<font color="Maroon" title="c19">v2</font>,<font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T104" title="GLIB_001:th.104">GLIB_001:104</a></i>;<br/><a NAME="E4:45_1_2_1_1_8_1_1_4_1_1_2_2"/><i><font color="Green" title="E74">A79</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1"/>
( <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v1</font> or <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font> )
 </span><b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_1_4"><i><font color="Green" title="E70">A74</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"/><i><font color="Green" title="E75">A80</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c20">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"><i><font color="Green" title="E75">A80</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c20">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_2"><i><font color="Green" title="E71">A76</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</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="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E74">A79</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"><i><font color="Green" title="E75">A80</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_1"/>
contradiction
 ; <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="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"/><i><font color="Green" title="E75">A81</font></i>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1"/>
( <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> or <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"><i><font color="Green" title="E75">A81</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E76">A82</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">n</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1"><i><font color="Green" title="E59">A52</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4"><i><font color="Green" title="E70">A74</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c9">n2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">n</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1"><i><font color="Green" title="E45">A25</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_2"><i><font color="Green" title="E76">A82</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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="45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c20">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>hence </b><a NAME="E2:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_1"><i><font color="Green" title="E44">A24</font></i></a>, <a class="txt" href="#E7:45_1_2_1_1"><i><font color="Green" title="E48">A28</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2_2"><i><font color="Green" title="E73">A78</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"><i><font color="Green" title="E75">A81</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</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><b>end;</b></div><b>hence </b><a NAME="E3:45_1_2_1_1_8_1_1_4_1_1_2_2_1_1_2"/>
contradiction
 ; <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><b>end;</b></div><b>hence </b><a NAME="E6:45_1_2_1_1_8_1_1_4_1_1_2_2"/>
contradiction
 ; <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="E5:45_1_2_1_1_8_1_1_4_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_1_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_1_4_1_1_2"><i><font color="Green" title="E72">A77</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1_4"><i><font color="Green" title="E69">A73</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XIN</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E5:45_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_1"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_1"><i><font color="Green" title="E60">A53</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c16">XOUT</font>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></i>
;<br/><b>hence </b><a NAME="E15:45_1_2_1_1_8_1_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E7:45_1_2_1_1_8_1_1"><i><font color="Green" title="E64">A57</font></i></a>, <a class="txt" href="#E11:45_1_2_1_1_8_1_1"><i><font color="Green" title="E66">A70</font></i></a>, <a class="txt" href="#E12:45_1_2_1_1_8_1_1"><i><font color="Green" title="E67">A71</font></i></a>, <a class="txt" href="#E13:45_1_2_1_1_8_1_1"><i><font color="Green" title="E68">A72</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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="45_1_2_1_1_8_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_2"/><i><font color="Green" title="E59">A83</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_2"/><b>then </b><i><font color="Green" title="E60">A84</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="#E11:45_1_2_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="#E14:45_1_2_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="txt" href="#E18:45_1_2_1_1"><i><font color="Green" title="E56">A40</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_2"/><i><font color="Green" title="E61">A85</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  )
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_2"><i><font color="Green" title="E59">A83</font></i></a>, <a class="ref" href="glib_000.html#T60" title="GLIB_000:th.60">GLIB_000:60</a></i>;<br/><b>set </b><font color="Maroon" title="c13">XIN1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">XIN2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><i><font color="Green" title="E62">A86</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c15">XIN1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><i><font color="Green" title="E63">A87</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">XIN1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_2"><i><font color="Green" title="E62">A86</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c16">XIN2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:45_1_2_1_1_8_1_2"/><i><font color="Green" title="E64">A88</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:45_1_2_1_1_8_1_2"/><i><font color="Green" title="E65">A89</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><div><i><font color="Green" title="E66">A90</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_2_3"/><i><font color="Green" title="E67">A91</font></i>: 
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_2_3"/><b>then </b><i><font color="Green" title="E68">A92</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_2_3"/><i><font color="Green" title="E69">A93</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_2"><i><font color="Green" title="E64">A88</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E67">A91</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_2_3_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or ( <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_2_3_1_1_1"/><i><font color="Green" title="E70">A94</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_2_3_1_1_1"/><b>then </b><i><font color="Green" title="E71">A95</font></i>: 
( not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E21:45_1_2_1_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_2"><i><font color="Green" title="E65">A89</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<font color="Maroon" title="c15">XIN1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2_3_1_1_1"><i><font color="Green" title="E71">A95</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3_1_1_1"><i><font color="Green" title="E70">A94</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_8_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_2"><i><font color="Green" title="E64">A88</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E67">A91</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="45_1_2_1_1_8_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_2_3_1_1_2"/><i><font color="Green" title="E70">A96</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_2_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:45_1_2_1_1_8_1_2"><i><font color="Green" title="E65">A89</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_2_3_1_1_2"><i><font color="Green" title="E70">A96</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2"><i><font color="Green" title="E60">A84</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3_1_1_2"><i><font color="Green" title="E70">A96</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_8_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_2"><i><font color="Green" title="E64">A88</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E67">A91</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="45_1_2_1_1_8_1_2_3_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_2_3_1_1_3"/><i><font color="Green" title="E70">A97</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_2_3_1_1_3"/><b>then </b><i><font color="Green" title="E71">A98</font></i>: 
( not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:45_1_2_1_1_8_1_2"><i><font color="Green" title="E65">A89</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E72">A99</font></i>: <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<font color="Maroon" title="c15">XIN1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2_3_1_1_3"><i><font color="Green" title="E71">A98</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E68">A92</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:45_1_2_1_1_8_1_2_3_1_1_3"/>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E68">A92</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_3_1_1_3"><i><font color="Green" title="E70">A97</font></i></a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_2_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E68">A92</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_2_3"><i><font color="Green" title="E69">A93</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_2_3_1_1_3"><i><font color="Green" title="E72">A99</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_2_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XIN2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 ; <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><a NAME="E11:45_1_2_1_1_8_1_2"/>
not <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:45_1_2_1_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_2"><i><font color="Green" title="E65">A89</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E67">A100</font></i>: <font color="Maroon" title="c15">XIN1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E13:45_1_2_1_1_8_1_2"/><i><font color="Green" title="E68">A101</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_2_5"/><i><font color="Green" title="E69">A102</font></i>: 
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_2_5"/><b>then </b><i><font color="Green" title="E70">A103</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_2_5"/><i><font color="Green" title="E71">A104</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )
 <b>by </b><i><a class="txt" href="#E13:45_1_2_1_1_8_1_2"><i><font color="Green" title="E68">A101</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E69">A102</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E4:45_1_2_1_1_8_1_2_5"/><i><font color="Green" title="E72">A105</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E70">A103</font></i></a>, <a class="ref" href="glib_000.html#T61" title="GLIB_000:th.61">GLIB_000:61</a></i>;<br/><a NAME="E5:45_1_2_1_1_8_1_2_5"/><b>then </b><i><font color="Green" title="E73">A106</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E15:45_1_2_1_1"><i><font color="Green" title="E53">A33</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2"><i><font color="Green" title="E59">A83</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E6:45_1_2_1_1_8_1_2_5"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E16:45_1_2_1_1"><i><font color="Green" title="E54">A35</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_2"><i><font color="Green" title="E59">A83</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E72">A105</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E7:45_1_2_1_1_8_1_2_5"/><b>then </b>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E70">A103</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E73">A106</font></i></a></i>;<br/><b>hence </b><a NAME="E8:45_1_2_1_1_8_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E70">A103</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_2_5"><i><font color="Green" title="E71">A104</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><a NAME="E15:45_1_2_1_1_8_1_2"/><b>then </b><i><font color="Green" title="E69">A107</font></i>: 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E13:45_1_2_1_1_8_1_2"><i><font color="Green" title="E68">A101</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XIN1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E12:45_1_2_1_1_8_1_2"><i><font color="Green" title="E67">A100</font></i></a>, <a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a>, <a class="txt" href="#E6:45_1_2_1_1_8_1_2"><i><font color="Green" title="E63">A87</font></i></a>, <a class="txt" href="#E8:45_1_2_1_1_8_1_2"><i><font color="Green" title="E64">A88</font></i></a>, <a class="txt" href="#E10:45_1_2_1_1_8_1_2"><i><font color="Green" title="E66">A90</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XIN1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_2"><i><font color="Green" title="E61">A85</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>

;<br/><b>hence </b><a NAME="E17:45_1_2_1_1_8_1_2"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A1a</font></i></a>, <a class="txt" href="#E4:45_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="#E15:45_1_2_1_1_8_1_2"><i><font color="Green" title="E69">A107</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</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="45_1_2_1_1_8_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_3"/><i><font color="Green" title="E59">A108</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_3"/><b>then </b><i><font color="Green" title="E60">A109</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="#E11:45_1_2_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="#E14:45_1_2_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="txt" href="#E17:45_1_2_1_1"><i><font color="Green" title="E55">A38</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_3"/><i><font color="Green" title="E61">A110</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_3"><i><font color="Green" title="E59">A108</font></i></a>, <a class="ref" href="glib_000.html#T62" title="GLIB_000:th.62">GLIB_000:62</a></i>;<br/><b>set </b><font color="Maroon" title="c13">XOUT1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">XOUT2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><i><font color="Green" title="E62">A111</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_3"><i><font color="Green" title="E61">A110</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c15">XOUT1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><i><font color="Green" title="E63">A112</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">XOUT1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_3"><i><font color="Green" title="E62">A111</font></i></a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_3"><i><font color="Green" title="E61">A110</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c16">XOUT2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E8:45_1_2_1_1_8_1_3"/><i><font color="Green" title="E64">A113</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:45_1_2_1_1_8_1_3"/><i><font color="Green" title="E65">A114</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><div><i><font color="Green" title="E66">A115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_3_3"/><i><font color="Green" title="E67">A116</font></i>: 
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_3_3"/><b>then </b><i><font color="Green" title="E68">A117</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_3_3"/><i><font color="Green" title="E69">A118</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_3"><i><font color="Green" title="E64">A113</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E67">A116</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_3_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_3_3_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or ( <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_3_3_1_1_1"/><i><font color="Green" title="E70">A119</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_3_3_1_1_1"/><b>then </b><i><font color="Green" title="E71">A120</font></i>: 
( not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E21:45_1_2_1_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_3"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<font color="Maroon" title="c15">XOUT1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3_3_1_1_1"><i><font color="Green" title="E71">A120</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3_1_1_1"><i><font color="Green" title="E70">A119</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_8_1_3_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_3"><i><font color="Green" title="E64">A113</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E67">A116</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="45_1_2_1_1_8_1_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_3_3_1_1_2"/><i><font color="Green" title="E70">A121</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_3_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:45_1_2_1_1_8_1_3"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3"><i><font color="Green" title="E60">A109</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3_1_1_2"><i><font color="Green" title="E70">A121</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><b>hence </b><a NAME="E4:45_1_2_1_1_8_1_3_3_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E8:45_1_2_1_1_8_1_3"><i><font color="Green" title="E64">A113</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E67">A116</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="45_1_2_1_1_8_1_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_3_3_1_1_3"/><i><font color="Green" title="E70">A122</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_3_3_1_1_3"/><b>then </b><i><font color="Green" title="E71">A123</font></i>: 
( not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> &amp; not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:45_1_2_1_1_8_1_3"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E72">A124</font></i>: <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<font color="Maroon" title="c15">XOUT1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3_3_1_1_3"><i><font color="Green" title="E71">A123</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E68">A117</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:45_1_2_1_1_8_1_3_3_1_1_3"/>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E68">A117</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_3_1_1_3"><i><font color="Green" title="E70">A122</font></i></a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E68">A117</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_3_3"><i><font color="Green" title="E69">A118</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_3_3_1_1_3"><i><font color="Green" title="E72">A124</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_3_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 ; <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><a NAME="E11:45_1_2_1_1_8_1_3"/>
not <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E21:45_1_2_1_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_3"><i><font color="Green" title="E65">A114</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E67">A125</font></i>: <font color="Maroon" title="c15">XOUT1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_3"><i><font color="Green" title="E61">A110</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><i><font color="Green" title="E68">A126</font></i>:  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XOUT1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E12:45_1_2_1_1_8_1_3"><i><font color="Green" title="E67">A125</font></i></a>, <a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a>, <a class="txt" href="#E6:45_1_2_1_1_8_1_3"><i><font color="Green" title="E63">A112</font></i></a>, <a class="txt" href="#E8:45_1_2_1_1_8_1_3"><i><font color="Green" title="E64">A113</font></i></a>, <a class="txt" href="#E10:45_1_2_1_1_8_1_3"><i><font color="Green" title="E66">A115</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <font color="Maroon" title="c15">XOUT1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_3"><i><font color="Green" title="E61">A110</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>

;<br/><a NAME="E14:45_1_2_1_1_8_1_3"/><i><font color="Green" title="E69">A127</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_3_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_3_6"/><i><font color="Green" title="E70">A128</font></i>: 
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_3_6"/><b>then </b><i><font color="Green" title="E71">A129</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_3_6"/><i><font color="Green" title="E72">A130</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )
 <b>by </b><i><a class="txt" href="#E14:45_1_2_1_1_8_1_3"><i><font color="Green" title="E69">A127</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E70">A128</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E4:45_1_2_1_1_8_1_3_6"/><i><font color="Green" title="E73">A131</font></i>: 
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> )
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E71">A129</font></i></a>, <a class="ref" href="glib_000.html#T59" title="GLIB_000:th.59">GLIB_000:59</a></i>;<br/><a NAME="E5:45_1_2_1_1_8_1_3_6"/><b>then </b><i><font color="Green" title="E74">A132</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E15:45_1_2_1_1"><i><font color="Green" title="E53">A33</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3"><i><font color="Green" title="E59">A108</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E6:45_1_2_1_1_8_1_3_6"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E16:45_1_2_1_1"><i><font color="Green" title="E54">A35</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_3"><i><font color="Green" title="E59">A108</font></i></a>, <a class="txt" href="#E4:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E73">A131</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E7:45_1_2_1_1_8_1_3_6"/><b>then </b>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E71">A129</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E74">A132</font></i></a></i>;<br/><b>hence </b><a NAME="E8:45_1_2_1_1_8_1_3_6"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E71">A129</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_3_6"><i><font color="Green" title="E72">A130</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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="E16:45_1_2_1_1_8_1_3"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:45_1"><i><font color="Green" title="E43">A23</font></i></a>, <a class="txt" href="#E13:45_1_2_1_1_8_1_3"><i><font color="Green" title="E68">A126</font></i></a>, <a class="txt" href="#E14:45_1_2_1_1_8_1_3"><i><font color="Green" title="E69">A127</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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="45_1_2_1_1_8_1_4"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_4"/><i><font color="Green" title="E59">A133</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">n2</font>,<font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_4"/><b>then </b><i><font color="Green" title="E60">A134</font></i>: 
( <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E5:45_1_2_1_1"><i><font color="Green" title="E46">A26</font></i></a>, <a class="txt" href="#E11:45_1_2_1_1"><i><font color="Green" title="E49">A29</font></i></a>, <a class="txt" href="#E14:45_1_2_1_1"><i><font color="Green" title="E52">A32</font></i></a>, <a class="txt" href="#E17:45_1_2_1_1"><i><font color="Green" title="E55">A38</font></i></a>, <a class="txt" href="#E18:45_1_2_1_1"><i><font color="Green" title="E56">A40</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_4"/><i><font color="Green" title="E61">A135</font></i>: 
( <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  &amp; <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  )
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_4"><i><font color="Green" title="E59">A133</font></i></a>, <a class="ref" href="glib_000.html#T60" title="GLIB_000:th.60">GLIB_000:60</a>, <a class="ref" href="glib_000.html#T62" title="GLIB_000:th.62">GLIB_000:62</a></i>;<br/><b>set </b><font color="Maroon" title="c13">XIN</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c14">XOUT</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E4:45_1_2_1_1_8_1_4"/><i><font color="Green" title="E62">A136</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then </b><i><font color="Green" title="E63">A137</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_4"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c15">XIN</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><i><font color="Green" title="E64">A138</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p0">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_4"><i><font color="Green" title="E62">A136</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_4"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></i>
;<br/><b>then reconsider </b><font color="Maroon" title="c16">XOUT</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> as   <a href="uproots.html#NM1" title="UPROOTS:NM.1">Rbag</a> of <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E9:45_1_2_1_1_8_1_4"/><i><font color="Green" title="E65">A139</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_4_3"/>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_4_3"/><b>then </b><i><font color="Green" title="E66">A140</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_4_3"/><b>then </b><i><font color="Green" title="E67">A141</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T59" title="GLIB_000:th.59">GLIB_000:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_4_3_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_3_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_4_3_1_1_1"/><i><font color="Green" title="E68">A142</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_4_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_4"><i><font color="Green" title="E62">A136</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_4_3_1_1_1"><i><font color="Green" title="E68">A142</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4"><i><font color="Green" title="E60">A134</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4_3_1_1_1"><i><font color="Green" title="E68">A142</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/> <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="45_1_2_1_1_8_1_4_3_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_4_3_1_1_2"/><i><font color="Green" title="E68">A143</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_4_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_4"><i><font color="Green" title="E62">A136</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E69">A144</font></i>: <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4_3"><i><font color="Green" title="E66">A140</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:45_1_2_1_1_8_1_4_3_1_1_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E15:45_1_2_1_1"><i><font color="Green" title="E53">A33</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4"><i><font color="Green" title="E59">A133</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_4_3"><i><font color="Green" title="E67">A141</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E5:45_1_2_1_1_8_1_4_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_4_3"><i><font color="Green" title="E66">A140</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4_3_1_1_2"><i><font color="Green" title="E68">A143</font></i></a></i>;<br/><b>hence </b><a NAME="E6:45_1_2_1_1_8_1_4_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_4_3"><i><font color="Green" title="E66">A140</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_4_3_1_1_2"><i><font color="Green" title="E69">A144</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_4_3"/>
<font color="Maroon" title="c15">XIN</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4_3"><i><font color="Green" title="E66">A140</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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>then </b><i><font color="Green" title="E66">A145</font></i>:  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a>, <a class="txt" href="#E5:45_1_2_1_1_8_1_4"><i><font color="Green" title="E63">A137</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_4"><i><font color="Green" title="E65">A139</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_4"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> implies <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_1_8_1_4_5"/>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><a NAME="E2:45_1_2_1_1_8_1_4_5"/><b>then </b><i><font color="Green" title="E67">A146</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_1_8_1_4_5"/><b>then </b><i><font color="Green" title="E68">A147</font></i>: 
<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T61" title="GLIB_000:th.61">GLIB_000:61</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_2_1_1_8_1_4_5_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2_1_1_8_1_4_5_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_4_5_1_1_1"/><i><font color="Green" title="E69">A148</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_4_5_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_4"><i><font color="Green" title="E62">A136</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:45_1_2_1_1_8_1_4_5_1_1_1"><i><font color="Green" title="E69">A148</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4"><i><font color="Green" title="E60">A134</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4_5_1_1_1"><i><font color="Green" title="E69">A148</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/> <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="45_1_2_1_1_8_1_4_5_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_1_8_1_4_5_1_1_2"/><i><font color="Green" title="E69">A149</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span><br/><div class="add"><a NAME="E2:45_1_2_1_1_8_1_4_5_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</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"><span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:45_1_2_1_1_8_1_4"><i><font color="Green" title="E62">A136</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E70">A150</font></i>: <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4_5"><i><font color="Green" title="E67">A146</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><a NAME="E4:45_1_2_1_1_8_1_4_5_1_1_2"/>
<font color="Maroon" title="c17">e</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="#E16:45_1_2_1_1"><i><font color="Green" title="E54">A35</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4"><i><font color="Green" title="E59">A133</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_4_5"><i><font color="Green" title="E68">A147</font></i></a>, <a class="ref" href="glib_000.html#D16" title="GLIB_000:def.16">GLIB_000:def 16</a></i>;<br/><a NAME="E5:45_1_2_1_1_8_1_4_5_1_1_2"/><b>then </b>
not <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 <b>by </b><i><a class="txt" href="#E22:45_1_2_1_1"><i><font color="Green" title="E58">A45</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_4_5"><i><font color="Green" title="E67">A146</font></i></a>, <a class="txt" href="#E1:45_1_2_1_1_8_1_4_5_1_1_2"><i><font color="Green" title="E69">A149</font></i></a></i>;<br/><b>hence </b><a NAME="E6:45_1_2_1_1_8_1_4_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_1_8_1_4_5"><i><font color="Green" title="E67">A146</font></i></a>, <a class="txt" href="#E3:45_1_2_1_1_8_1_4_5_1_1_2"><i><font color="Green" title="E70">A150</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><b>end;</b></div><b>hence </b><a NAME="E5:45_1_2_1_1_8_1_4_5"/>
<font color="Maroon" title="c16">XOUT</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_1_8_1_4_5"><i><font color="Green" title="E67">A146</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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>then </b> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a>, <a class="txt" href="#E7:45_1_2_1_1_8_1_4"><i><font color="Green" title="E64">A138</font></i></a>, <a class="txt" href="#E9:45_1_2_1_1_8_1_4"><i><font color="Green" title="E65">A139</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">n2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E3:45_1_2_1_1_8_1_4"><i><font color="Green" title="E61">A135</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:45_1"><i><font color="Green" title="E43">A23</font></i></a></i>
;<br/><b>hence </b><a NAME="E14:45_1_2_1_1_8_1_4"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:45_1_2_1_1_8_1_4"><i><font color="Green" title="E66">A145</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><b>end;</b></div><b>hence </b><a NAME="E24:45_1_2_1_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <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="45_1_2_1_2"><b>suppose </b></a><a NAME="E1:45_1_2_1_2"/><i><font color="Green" title="E44">A151</font></i>: 
not <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:45_1_2_1_2"/><i><font color="Green" title="E45">A152</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_2"/><b>then </b><i><font color="Green" title="E46">A153</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green" title="E47">A158a</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">e</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"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_2_1"/><i><font color="Green" title="E48">A154</font></i>: 
<font color="Maroon" title="c8">e</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"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font></span><br/><a NAME="E2:45_1_2_1_2_1"/><b>then </b><i><font color="Green" title="E49">A155</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> )
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_2"><i><font color="Green" title="E45">A152</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_2_1_1"/><i><font color="Green" title="E50">A156</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_2_1_1"/><i><font color="Green" title="E51">A157</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_2"><i><font color="Green" title="E45">A152</font></i></a>, <a class="txt" href="#E1:45_1_2_1_2_1"><i><font color="Green" title="E48">A154</font></i></a>, <a class="ref" href="glib_000.html#T60" title="GLIB_000:th.60">GLIB_000:60</a></i>;<br/><a NAME="E4:45_1_2_1_2_1_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_2_1_1"><i><font color="Green" title="E51">A157</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_2"><i><font color="Green" title="E44">A151</font></i></a>, <a class="txt" href="#E1:45_1_2_1_2_1_1"><i><font color="Green" title="E50">A156</font></i></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</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="E4:45_1_2_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_2"><i><font color="Green" title="E45">A152</font></i></a>, <a class="txt" href="#E1:45_1_2_1_2_1"><i><font color="Green" title="E48">A154</font></i></a>, <a class="txt" href="#E2:45_1_2_1_2_1"><i><font color="Green" title="E49">A155</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><a NAME="E5:45_1_2_1_2"/><i><font color="Green" title="E48">A159</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  )
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">e</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"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> )</span><br/><b>assume </b><a NAME="E1:45_1_2_1_2_2"/><i><font color="Green" title="E49">A160</font></i>: 
<font color="Maroon" title="c8">e</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"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font></span><br/><a NAME="E2:45_1_2_1_2_2"/><b>then </b><i><font color="Green" title="E50">A161</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:45_1_2_1_2_2"/><i><font color="Green" title="E51">A162</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> )
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_2"><i><font color="Green" title="E48">A159</font></i></a>, <a class="txt" href="#E1:45_1_2_1_2_2"><i><font color="Green" title="E49">A160</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1_2_1_2_2_1"/><i><font color="Green" title="E52">A163</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">P</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:45_1_2_1_2_2_1"/><i><font color="Green" title="E53">A164</font></i>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E2:45_1_2_1_2_2"><i><font color="Green" title="E50">A161</font></i></a>, <a class="ref" href="glib_000.html#T62" title="GLIB_000:th.62">GLIB_000:62</a></i>;<br/><a NAME="E4:45_1_2_1_2_2_1"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">v</font>,<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E3:45_1_2_1_2_2_1"><i><font color="Green" title="E53">A164</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><b>hence </b><a NAME="E5:45_1_2_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:45_1_2_1_2"><i><font color="Green" title="E44">A151</font></i></a>, <a class="txt" href="#E1:45_1_2_1_2_2_1"><i><font color="Green" title="E52">A163</font></i></a>, <a class="ref" href="glib_001.html#T106" title="GLIB_001:th.106">GLIB_001:106</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="E5:45_1_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c8">e</font>
 <b>by </b><i><a class="txt" href="#E4:45"><i><font color="Green" title="E41">A1c</font></i></a>, <a class="txt" href="#E2:45_1_2_1_2_2"><i><font color="Green" title="E50">A161</font></i></a>, <a class="txt" href="#E3:45_1_2_1_2_2"><i><font color="Green" title="E51">A162</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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><a NAME="E7:45_1_2_1_2"/><b>then </b><i><font color="Green" title="E49">A165</font></i>: 
<font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="#E5:45_1_2_1_2"><i><font color="Green" title="E48">A159</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>thus </b> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> = 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:45_1_2_1_2"><i><font color="Green" title="E47">A158a</font></i></a>, <a class="txt" href="#E3:45_1_2_1_2"><i><font color="Green" title="E46">A153</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>
<br/>.= 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E7:45_1_2_1_2"><i><font color="Green" title="E49">A165</font></i></a>, <a class="txt" href="#E2:45"><i><font color="Green" title="E39">A1a</font></i></a>, <a class="txt" href="#E4:45_1"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</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><b>end;</b></div><b>hence </b><a NAME="E7:45_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span></span>)</span>
 ; <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="E6:45"/>
 <a href="glib_005.html#K10" title="GLIB_005:func.10">FF:PushFlow</a> <font color="Maroon" title="c2">EL</font>,<font color="Maroon" title="c5">P</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font>
 <b>by </b><i><a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</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>
