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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/><span class="p1">(<span class="default"><font color="Olive" title="b1">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Olive" title="b1">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="Olive" title="b1">EL</font>,<font color="Olive" title="b4">P</font></span>)</span> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</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/><span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">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="Olive" title="b3">P</font></span>)</span> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</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/><span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">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="Olive" title="b1">P</font></span>)</span> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</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 <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="glib_005.html#K1" title="GLIB_005:func.1">.flow</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="E1:46"/><i><font color="Green" title="E39">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="E2:46"/><i><font color="Green" title="E40">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="E3:46"/><i><font color="Green" title="E41">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="E4:46"/><i><font color="Green" title="E42">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="glib_005.html#K1" title="GLIB_005:func.1">.flow</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/>
<b>set </b><font color="Maroon" title="c7">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/>
<b>set </b><font color="Maroon" title="c8">EI1</font> = <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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">EO1</font> = <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<a NAME="E5:46"/>
( <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="#E3:46"><i><font color="Green" title="E41">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="E6:46"/><b>then </b>
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="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="glib_001.html#T128" title="GLIB_001:th.128">GLIB_001:128</a></i>;<br/>
<a NAME="E7:46"/><b>then </b>
3 <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="ref" href="glib_001.html#T126" title="GLIB_001:th.126">GLIB_001:126</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">lenP2g</font> = <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> <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>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<a NAME="E9:46"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c10">lenP2g</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="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/>
<a NAME="E10:46"/><b>then </b><i><font color="Green" title="E44">A4</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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</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="E11:46"/><b>then </b><i><font color="Green" title="E45">A5</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="c10">lenP2g</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="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#D15" title="GLIB_000:def.15">GLIB_000:def 15</a></i>;<br/>
<a NAME="E12:46"/><i><font color="Green" title="E46">A7</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></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="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> or  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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>,<font color="Maroon" title="c1">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1"/><i><font color="Green" title="E47">A8</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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</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"> <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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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></span><br/><div class="add"><a NAME="E2:46_1_1_1"/><b>then </b><i><font color="Green" title="E48">A9</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="c10">lenP2g</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="c10">lenP2g</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="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E9:46"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><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> <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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<b>by </b><i><a class="txt" href="#E1:46_1_1_1"><i><font color="Green" title="E47">A8</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="c4">sink</font>
<b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>
;<br/><a NAME="E4:46_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> <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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E5:46_1_1_1"/><b>then </b><i><font color="Green" title="E49">A10</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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E11:46"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="glib_000.html#D28" title="GLIB_000:def.28">GLIB_000:def 28</a></i>;<br/><b>set </b><font color="Maroon" title="c12">EI2</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="E50">A11</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E5:46_1_1_1"><i><font color="Green" title="E49">A10</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="c13">EI2</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E8:46_1_1_1"/><i><font color="Green" title="E51">A12</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><b>assume </b><a NAME="E1:46_1_1_1_3"/><i><font color="Green" title="E52">A13</font></i>: 
<font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:46_1_1_1_3"/><b>then </b><i><font color="Green" title="E53">A14</font></i>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E3:46_1_1_1_3"/><i><font color="Green" title="E54">A15</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">e</font>
 <b>by </b><i><a class="txt" href="#E8:46_1_1_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="txt" href="#E1:46_1_1_1_3"><i><font color="Green" title="E52">A13</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E4:46_1_1_1_3"/>
<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="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:46_1_1_1_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="ref" href="glib_000.html#D28" title="GLIB_000:def.28">GLIB_000:def 28</a></i>;<br/><a NAME="E5:46_1_1_1_3"/><b>then </b><i><font color="Green" title="E55">A16</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="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_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="E1:46_1_1_1_3_1_1"/>
( <font color="Maroon" title="c14">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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c14">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="c10">lenP2g</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="46_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1_3_1_1_1"/><i><font color="Green" title="E56">A17</font></i>: 
<font color="Maroon" title="c14">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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:46_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c10">lenP2g</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:46_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then </b><font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c10">lenP2g</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="#E1:46_1_1_1_3_1_1_1"><i><font color="Green" title="E56">A17</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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> <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="c10">lenP2g</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="#E2:46_1_1_1"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="#E2:46_1_1_1_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="#E1:46_1_1_1_3_1_1_1"><i><font color="Green" title="E56">A17</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E5:46_1_1_1_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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E8:46_1_1_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="txt" href="#E1:46_1_1_1_3"><i><font color="Green" title="E52">A13</font></i></a>, <a class="txt" href="#E1:46_1_1_1_3_1_1_1"><i><font color="Green" title="E56">A17</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="46_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_1_3_1_1_2"/><i><font color="Green" title="E56">A18</font></i>: 
<font color="Maroon" title="c14">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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:46_1_1_1_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c10">lenP2g</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:46_1_1_1_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="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="E57">A19</font></i>: <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">e</font>
<b>by </b><i><a class="txt" href="#E2:46_1_1_1_3"><i><font color="Green" title="E53">A14</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="46_1_1_1_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_1_3_1_1_2_2"/>
<font color="Maroon" title="c14">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="c15">v1</font>, <font color="Maroon" title="c16">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="c17">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="E3:46_1_1_1_3_1_1_2_2"/><i><font color="Green" title="E58">A20</font></i>: 
( <font color="Maroon" title="c17">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="c15">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="c17">m</font> &amp; <font color="Maroon" title="c14">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_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="46_1_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="E1:46_1_1_1_3_1_1_2_2_1_1"/>
( <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> or <font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 </span><b>by </b><i><a class="txt" href="#E5:46_1_1_1_3"><i><font color="Green" title="E55">A16</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</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="46_1_1_1_3_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1_3_1_1_2_2_1_1_1"/><i><font color="Green" title="E59">A21</font></i>: 
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_1_3_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E60">A22</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="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E3:46_1_1_1_3_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:46_1_1_1_3_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_1_3_1_1_2_2_1_1_1"><i><font color="Green" title="E60">A22</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_1_3_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="#E1:46_1_1_1_3_1_1_2_2_1_1_1"><i><font color="Green" title="E59">A21</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="46_1_1_1_3_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_1_3_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_1_3_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E59">A23</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="c17">m</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> <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>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_3_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_1_3_1_1_2_2_1_1_2_1"/>
<font color="Maroon" title="c17">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>  <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/><a NAME="E2:46_1_1_1_3_1_1_2_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">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> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_1_3_1_1_2_2_1_1_2"><i><font color="Green" title="E59">A23</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="E3:46_1_1_1_3_1_1_2_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 ;<br/><b>hence </b><a NAME="E4:46_1_1_1_3_1_1_2_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</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="E4:46_1_1_1_3_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">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>  <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="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_1_3_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46_1_1_1_3_1_1_2"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3_1_1_2_2"><i><font color="Green" title="E58">A20</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="E5:46_1_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="E6:46_1_1_1_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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E2:46_1_1_1_3"><i><font color="Green" title="E53">A14</font></i></a>, <a class="txt" href="#E3:46_1_1_1_3"><i><font color="Green" title="E54">A15</font></i></a>, <a class="txt" href="#E4:46_1_1_1_3_1_1_2"><i><font color="Green" title="E57">A19</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="E7:46_1_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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EI2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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>then </b><i><font color="Green" title="E52">A24</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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="#E6:46_1_1_1"><i><font color="Green" title="E50">A11</font></i></a>, <a class="txt" href="#E8:46_1_1_1"><i><font color="Green" title="E51">A12</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> <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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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/><a NAME="E11:46_1_1_1"/><i><font color="Green" title="E53">A25</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><b>assume </b><a NAME="E1:46_1_1_1_5"/><i><font color="Green" title="E54">A26</font></i>: 
<font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:46_1_1_1_5"/><b>then </b><i><font color="Green" title="E55">A27</font></i>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E3:46_1_1_1_5"/><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="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#D29" title="GLIB_000:def.29">GLIB_000:def 29</a></i>;<br/><a NAME="E4:46_1_1_1_5"/><b>then </b><i><font color="Green" title="E56">A28</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="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_1_5_1"/>
<font color="Maroon" title="c14">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="c15">v1</font>, <font color="Maroon" title="c16">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="c17">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="E3:46_1_1_1_5_1"/><i><font color="Green" title="E57">A29</font></i>: 
( <font color="Maroon" title="c17">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="c15">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="c17">m</font> &amp; <font color="Maroon" title="c14">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_1_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_1_5_1_1_1"/>
( <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> or <font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 </span><b>by </b><i><a class="txt" href="#E4:46_1_1_1_5"><i><font color="Green" title="E56">A28</font></i></a>, <a class="txt" href="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</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="46_1_1_1_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_1_5_1_1_1_1"/><i><font color="Green" title="E58">A30</font></i>: 
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_1_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E59">A31</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="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E3:46_1_1_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:46_1_1_1_5_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_1_5_1_1_1_1"><i><font color="Green" title="E59">A31</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_1_5_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="txt" href="#E1:46_1_1_1_5_1_1_1_1"><i><font color="Green" title="E58">A30</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="46_1_1_1_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_1_5_1_1_1_2"/>
<font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E58">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="c17">m</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> <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>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_1_5_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_1_5_1_1_1_2_1"/>
<font color="Maroon" title="c17">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>  <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/><a NAME="E2:46_1_1_1_5_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">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> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_1_5_1_1_1_2"><i><font color="Green" title="E58">A32</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="E3:46_1_1_1_5_1_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1_1_5_1_1_1_2_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><a NAME="E4:46_1_1_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">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>  <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="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:46_1_1_1_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A33</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="c10">lenP2g</font> <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="#E1:46_1_1_1"><i><font color="Green" title="E47">A8</font></i></a>, <a class="txt" href="#E4:46_1_1_1_5"><i><font color="Green" title="E56">A28</font></i></a>, <a class="txt" href="#E3:46_1_1_1_5_1"><i><font color="Green" title="E57">A29</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="E6:46_1_1_1_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E9:46"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E12:46"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E7:46_1_1_1_5_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E5:46_1_1_1_5_1_1_1_2"><i><font color="Green" title="E59">A33</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="E5:46_1_1_1_5_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>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="c14">e</font> = 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<b>by </b><i><a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E2:46_1_1_1_5"><i><font color="Green" title="E55">A27</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<b>by </b><i><a class="txt" href="#E2:46_1_1_1_5"><i><font color="Green" title="E55">A27</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E7:46_1_1_1_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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E11:46_1_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="txt" href="#E1:46_1_1_1_5"><i><font color="Green" title="E54">A26</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="E13:46_1_1_1"/><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E11:46_1_1_1"><i><font color="Green" title="E53">A25</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:46"><i><font color="Green" title="E40">A1a</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E10:46_1_1_1"><i><font color="Green" title="E52">A24</font></i></a>, <a class="ref" href="glib_005.html#D3" target="_self" title="GLIB_005:def.3">Def8</a>, <a class="ref" href="glib_005.html#T14" target="_self" title="GLIB_005:th.14">Th15</a></i>
<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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="#E2:46"><i><font color="Green" title="E40">A1a</font></i></a>, <a class="ref" href="glib_005.html#D3" target="_self" title="GLIB_005:def.3">Def8</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="46_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2"/><i><font color="Green" title="E47">A34</font></i>: 
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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</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"> <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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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></span><br/><div class="add"><a NAME="E2:46_1_1_2"/><b>then </b><i><font color="Green" title="E48">A35</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="c10">lenP2g</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="c10">lenP2g</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="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E9:46"><i><font color="Green" title="E43">A3</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</a></i>;<br/><a NAME="E3:46_1_1_2"/><i><font color="Green" title="E49">A36</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="c10">lenP2g</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="c10">lenP2g</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="c10">lenP2g</font>,<font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="#E10:46"><i><font color="Green" title="E44">A4</font></i></a>, <a class="txt" href="#E1:46_1_1_2"><i><font color="Green" title="E47">A34</font></i></a>, <a class="ref" href="glib_000.html#T19" title="GLIB_000:th.19">GLIB_000:19</a></i>;<br/><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> <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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> = 
<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="c10">lenP2g</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/>.= 
<font color="Maroon" title="c4">sink</font>
<b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>
;<br/><a NAME="E5:46_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> <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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:46_1_1_2"/><b>then </b><i><font color="Green" title="E50">A37</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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E11:46"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="glib_000.html#D29" title="GLIB_000:def.29">GLIB_000:def 29</a></i>;<br/><b>set </b><font color="Maroon" title="c12">EO2</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="E51">A38</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E6:46_1_1_2"><i><font color="Green" title="E50">A37</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="c13">EO2</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E9:46_1_1_2"/><i><font color="Green" title="E52">A39</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><b>assume </b><a NAME="E1:46_1_1_2_3"/><i><font color="Green" title="E53">A40</font></i>: 
<font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:46_1_1_2_3"/><b>then </b><i><font color="Green" title="E54">A41</font></i>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E3:46_1_1_2_3"/><i><font color="Green" title="E55">A42</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">e</font>
 <b>by </b><i><a class="txt" href="#E9:46_1_1_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="#E1:46_1_1_2_3"><i><font color="Green" title="E53">A40</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E4:46_1_1_2_3"/>
<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="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="#E2:46_1_1_2_3"><i><font color="Green" title="E54">A41</font></i></a>, <a class="ref" href="glib_000.html#D29" title="GLIB_000:def.29">GLIB_000:def 29</a></i>;<br/><a NAME="E5:46_1_1_2_3"/><b>then </b><i><font color="Green" title="E56">A43</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="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_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="E1:46_1_1_2_3_1_1"/>
( <font color="Maroon" title="c14">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="c10">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> or <font color="Maroon" title="c14">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="c10">lenP2g</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="46_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_2_3_1_1_1"/><i><font color="Green" title="E57">A44</font></i>: 
<font color="Maroon" title="c14">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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:46_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c10">lenP2g</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:46_1_1_2_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><b>then </b><font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c10">lenP2g</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="#E1:46_1_1_2_3_1_1_1"><i><font color="Green" title="E57">A44</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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="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="c14">e</font>
<b>by </b><i><a class="txt" href="#E2:46_1_1_2"><i><font color="Green" title="E48">A35</font></i></a>, <a class="txt" href="#E2:46_1_1_2_3"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="#E1:46_1_1_2_3_1_1_1"><i><font color="Green" title="E57">A44</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E5:46_1_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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E9:46_1_1_2"><i><font color="Green" title="E52">A39</font></i></a>, <a class="txt" href="#E1:46_1_1_2_3"><i><font color="Green" title="E53">A40</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="46_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2_3_1_1_2"/><i><font color="Green" title="E57">A45</font></i>: 
<font color="Maroon" title="c14">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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:46_1_1_2_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c10">lenP2g</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="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:46_1_1_2_3_1_1_2"/><b>then </b>
not <font color="Maroon" title="c14">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="c10">lenP2g</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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="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="E58">A46</font></i>: <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c14">e</font>
<b>by </b><i><a class="txt" href="#E2:46_1_1_2_3"><i><font color="Green" title="E54">A41</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="46_1_1_2_3_1_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2_3_1_1_2_2"/>
<font color="Maroon" title="c14">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="c15">v1</font>, <font color="Maroon" title="c16">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="c17">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="E3:46_1_1_2_3_1_1_2_2"/><i><font color="Green" title="E59">A47</font></i>: 
( <font color="Maroon" title="c17">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="c15">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="c17">m</font> &amp; <font color="Maroon" title="c14">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_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="E1:46_1_1_2_3_1_1_2_2_1_1"/>
( <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> or <font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 </span><b>by </b><i><a class="txt" href="#E5:46_1_1_2_3"><i><font color="Green" title="E56">A43</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</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="46_1_1_2_3_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_2_3_1_1_2_2_1_1_1"/><i><font color="Green" title="E60">A48</font></i>: 
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_2_3_1_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E61">A49</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="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E3:46_1_1_2_3_1_1_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:46_1_1_2_3_1_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_2_3_1_1_2_2_1_1_1"><i><font color="Green" title="E61">A49</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_2_3_1_1_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="txt" href="#E1:46_1_1_2_3_1_1_2_2_1_1_1"><i><font color="Green" title="E60">A48</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="46_1_1_2_3_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2_3_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_2_3_1_1_2_2_1_1_2"/><b>then </b><i><font color="Green" title="E60">A50</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="c17">m</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> <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>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_3_1_1_2_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2_3_1_1_2_2_1_1_2_1"/>
<font color="Maroon" title="c17">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>  <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/><a NAME="E2:46_1_1_2_3_1_1_2_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">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> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_2_3_1_1_2_2_1_1_2"><i><font color="Green" title="E60">A50</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="E3:46_1_1_2_3_1_1_2_2_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1_2_3_1_1_2_2_1_1_2_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><a NAME="E4:46_1_1_2_3_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">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>  <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="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_2_3_1_1_2_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46_1_1_2_3_1_1_2"><i><font color="Green" title="E57">A45</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3_1_1_2_2"><i><font color="Green" title="E59">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><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:46_1_1_2_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="E6:46_1_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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E2:46_1_1_2_3"><i><font color="Green" title="E54">A41</font></i></a>, <a class="txt" href="#E3:46_1_1_2_3"><i><font color="Green" title="E55">A42</font></i></a>, <a class="txt" href="#E4:46_1_1_2_3_1_1_2"><i><font color="Green" title="E58">A46</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="E7:46_1_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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">EO2</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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>then </b><i><font color="Green" title="E53">A51</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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"><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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p0"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <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="c10">lenP2g</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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c10">lenP2g</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="#E7:46_1_1_2"><i><font color="Green" title="E51">A38</font></i></a>, <a class="txt" href="#E9:46_1_1_2"><i><font color="Green" title="E52">A39</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> <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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/><a NAME="E12:46_1_1_2"/><i><font color="Green" title="E54">A52</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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"><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">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="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><b>assume </b><a NAME="E1:46_1_1_2_5"/><i><font color="Green" title="E55">A53</font></i>: 
<font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:46_1_1_2_5"/><b>then </b><i><font color="Green" title="E56">A54</font></i>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></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/><a NAME="E3:46_1_1_2_5"/><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="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="glib_000.html#D28" title="GLIB_000:def.28">GLIB_000:def 28</a></i>;<br/><a NAME="E4:46_1_1_2_5"/><b>then </b><i><font color="Green" title="E57">A55</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="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2_5_1"/>
<font color="Maroon" title="c14">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="c15">v1</font>, <font color="Maroon" title="c16">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="c17">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="E3:46_1_1_2_5_1"/><i><font color="Green" title="E58">A56</font></i>: 
( <font color="Maroon" title="c17">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="c15">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="c17">m</font> &amp; <font color="Maroon" title="c14">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">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="c17">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> &amp; <font color="Maroon" title="c14">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c15">v1</font>,<font color="Maroon" title="c16">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_5_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1_2_5_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1_1_2_5_1_1_1"/>
( <font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> or <font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font> )
 </span><b>by </b><i><a class="txt" href="#E4:46_1_1_2_5"><i><font color="Green" title="E57">A55</font></i></a>, <a class="txt" href="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</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="46_1_1_2_5_1_1_1_1"><b>suppose </b></a><a NAME="E1:46_1_1_2_5_1_1_1_1"/><i><font color="Green" title="E59">A57</font></i>: 
<font color="Maroon" title="c15">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_2_5_1_1_1_1"/><b>then </b><i><font color="Green" title="E60">A58</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="c17">m</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"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><a NAME="E3:46_1_1_2_5_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">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="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="xreal_1.html#T17" title="XREAL_1:th.17">XREAL_1:17</a></i>;<br/><a NAME="E4:46_1_1_2_5_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_2_5_1_1_1_1"><i><font color="Green" title="E60">A58</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E5:46_1_1_2_5_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="txt" href="#E1:46_1_1_2_5_1_1_1_1"><i><font color="Green" title="E59">A57</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="46_1_1_2_5_1_1_1_2"><b>suppose </b></a><a NAME="E1:46_1_1_2_5_1_1_1_2"/>
<font color="Maroon" title="c16">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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"> contradiction</span><br/><div class="add"><a NAME="E2:46_1_1_2_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E59">A59</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="c17">m</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> <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>
 <b>by </b><i><a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1_2_5_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:46_1_1_2_5_1_1_1_2_1"/>
<font color="Maroon" title="c17">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>  <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/><a NAME="E2:46_1_1_2_5_1_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">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> 1
 <b>by </b><i><a class="txt" href="#E2:46_1_1_2_5_1_1_1_2"><i><font color="Green" title="E59">A59</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="E3:46_1_1_2_5_1_1_1_2_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 2
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><b>hence </b><a NAME="E4:46_1_1_2_5_1_1_1_2_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><a NAME="E4:46_1_1_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c17">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>  <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="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E5:46_1_1_2_5_1_1_1_2"/><b>then </b><i><font color="Green" title="E60">A60</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="c10">lenP2g</font> <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="#E3:46_1_1_2"><i><font color="Green" title="E49">A36</font></i></a>, <a class="txt" href="#E4:46_1_1_2_5"><i><font color="Green" title="E57">A55</font></i></a>, <a class="txt" href="#E3:46_1_1_2_5_1"><i><font color="Green" title="E58">A56</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="E6:46_1_1_2_5_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c10">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="#E9:46"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E12:46"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></i>;<br/><b>hence </b><a NAME="E7:46_1_1_2_5_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E5:46_1_1_2_5_1_1_1_2"><i><font color="Green" title="E60">A60</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</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="E5:46_1_1_2_5_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>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="c14">e</font> = 
<font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<b>by </b><i><a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E2:46_1_1_2_5"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def20</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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<b>by </b><i><a class="txt" href="#E2:46_1_1_2_5"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/><b>hence </b><a NAME="E7:46_1_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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <b>by </b><i><a class="txt" href="#E12:46_1_1_2"><i><font color="Green" title="E54">A52</font></i></a>, <a class="txt" href="#E1:46_1_1_2_5"><i><font color="Green" title="E55">A53</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="E14:46_1_1_2"/><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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E12:46_1_1_2"><i><font color="Green" title="E54">A52</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> = 
<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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>
<b>by </b><i><a class="txt" href="#E1:46"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E2:46"><i><font color="Green" title="E40">A1a</font></i></a>, <a class="txt" href="#E3:46"><i><font color="Green" title="E41">A1b</font></i></a>, <a class="txt" href="#E4:46"><i><font color="Green" title="E42">A1c</font></i></a>, <a class="txt" href="#E11:46_1_1_2"><i><font color="Green" title="E53">A51</font></i></a>, <a class="ref" href="glib_005.html#D3" target="_self" title="GLIB_005:def.3">Def8</a>, <a class="ref" href="glib_005.html#T14" target="_self" title="GLIB_005:th.14">Th15</a></i>
<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="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.edgesInto</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <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="c1">G</font> <a href="glib_000.html#K20" title="GLIB_000:func.20">.edgesOutOf</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">sink</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="#E2:46"><i><font color="Green" title="E40">A1a</font></i></a>, <a class="ref" href="glib_005.html#D3" target="_self" title="GLIB_005:def.3">Def8</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><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E14:46"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<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"> verum</span><br/>


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