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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="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/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E4:48"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E5:48"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></span>;<br/>
<a NAME="E6:48"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">P</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="glib_001.html#T127" title="GLIB_001:th.127">GLIB_001:127</a></span>;<br/>
<a NAME="E7:48"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T125" title="GLIB_001:th.125">GLIB_001:125</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T5" title="INT_1:th.5">INT_1:5</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">EI1</font> = <font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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/>
<span class="kw">set </span><font color="Maroon" title="c9">EO1</font> = <font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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/>
<span class="kw">set </span><font color="Maroon" title="c10">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/>
<span class="kw">set </span><font color="Maroon" title="c11">T</font> = <font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font>;<br/>
<a NAME="E9:48"/><span class="lab"><font color="Green" title="E42">A5</font></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"><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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/>
<a NAME="E10:48"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c6">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/>
<a NAME="E11:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A7</font></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="c6">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="c6">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="c6">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 class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#D3" title="GLIB_001:def.3">GLIB_001:def 3</a></span>;<br/>
<a NAME="E12:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A8</font></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="c6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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="c6">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="c6">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="c6">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="c6">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="c6">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="c6">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="48_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1"/><span class="lab"><font color="Green" title="E46">A9</font></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="c6">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="c6">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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">then </span><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="c6">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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c4">sink</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>
;<br/><a NAME="E3:48_1_1_1"/><span class="kw">then </span>
<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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A10</font></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="c6">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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">EI2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E5:48_1_1_1"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E49">A12</font></span>:  <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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">EI2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><a NAME="E8:48_1_1_1"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<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="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="c6">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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_1_1_3"/><span class="lab"><font color="Green" title="E51">A14</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:48_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:48_1_1_1_3"/><span class="lab"><font color="Green" title="E53">A16</font></span>: 
<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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><a NAME="E4:48_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_3"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><a NAME="E5:48_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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="c6">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="c6">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="48_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
<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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:48_1_1_1_3_1_1_1"/><span class="kw">then </span>
<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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:48_1_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
<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="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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_1_1_1"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E2:48_1_1_1_3"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A19</font></span>: 
<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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E56">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_1_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A21</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A22</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:48_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A23</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:48_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A24</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_1_3_1_1_2_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_1_3"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E7:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E61">A25</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A26</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A27</font></span>: 
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E3:48_1_1_1_3_1_1_2_1_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E4:48_1_1_1_3_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E63">A27</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_1_3_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A22</font></span></a>, <a class="txt" href="#E1:48_1_1_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A26</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_1_3_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A28</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E6:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E60">A24</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_3_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_1_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_3_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E62">A28</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:48_1_1_1_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 ;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_1_3_1_1_2_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:48_1_1_1_3_1_1_2_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E57">A21</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_1_3_1_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="txt" href="#E5:48_1_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E59">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:48_1_1_1_3_1_1_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:48_1_1_1_3_1_1_2"/>
 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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_1_1_3_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_3"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E2:48_1_1_1_3"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E3:48_1_1_1_3"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E2:48_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A20</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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="c13">EI2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A29</font></span>:  <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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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"><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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="funct_1.html#K1" title="FUNCT_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="c6">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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_1"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E5:48_1_1_1"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a>, <a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></span>
<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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="E11:48_1_1_1"/><span class="lab"><font color="Green" title="E52">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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#K19" title="GLIB_000:func.19">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><div><span class="lab"><font color="Green" title="E53">A31</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_1_1_5"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:48_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A33</font></span>: 
<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">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:48_1_1_1_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><a NAME="E4:48_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_1_5_1"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_1_5_1"/><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:48_1_1_1_5_1"/><span class="lab"><font color="Green" title="E59">A37</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:48_1_1_1_5_1"/><span class="lab"><font color="Green" title="E60">A38</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_1_5_1"/><span class="lab"><font color="Green" title="E61">A39</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_5_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_5_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_1_5"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E7:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E61">A39</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A40</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_1_5_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A41</font></span>: 
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E3:48_1_1_1_5_1_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E1:48_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E62">A40</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E4:48_1_1_1_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E63">A41</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_1_5_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="txt" href="#E1:48_1_1_1_5_1_1_1_1"><span class="lab"><font color="Green" title="E62">A40</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_1_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A42</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E6:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_1_5_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_1_5_1_1_1_2_1"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E62">A42</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:48_1_1_1_5_1_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_1_5_1_1_1_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:48_1_1_1_5_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:48_1_1_1_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A43</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_1"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E4:48_1_1_1_5"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E5:48_1_1_1_5_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>;<br/><a NAME="E6:48_1_1_1_5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_1_1_5_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E5:48_1_1_1_5_1_1_1_2"><span class="lab"><font color="Green" title="E63">A43</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:48_1_1_1_5_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> = 
<font color="Maroon" title="c2">EL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E2:48_1_1_1_5"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_1_5"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E1:48_1_1_1_5"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E13:48_1_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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#K19" title="GLIB_000:func.19">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="kw">hence </span><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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48_1_1_1"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="txt" href="#E11:48_1_1_1"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E12:48_1_1_1"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2"/><span class="lab"><font color="Green" title="E46">A44</font></span>: 
 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="c6">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="c6">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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E2:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A45</font></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="c6">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="c6">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="c6">lenP2g</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="glib_000.html#T16" title="GLIB_000:th.16">GLIB_000:16</a></span>;<br/><span class="kw">then </span><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="c6">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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c4">sink</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>
;<br/><a NAME="E4:48_1_1_2"/><span class="kw">then </span>
<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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E5:48_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A46</font></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="c6">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">.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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c12">EO2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="glib_005.html#K9" title="GLIB_005:func.9">.tolerance</a> <font color="Maroon" title="c2">EL</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E6:48_1_1_2"/><span class="lab"><font color="Green" title="E49">A47</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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#K19" title="GLIB_000:func.19">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="lab"><font color="Green" title="E50">A48</font></span>:  <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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<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">.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="c6">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K19" title="GLIB_000:func.19">.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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_2"><span class="lab"><font color="Green" title="E48">A46</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" title="ZFMISC_1:th.40">ZFMISC_1:40</a></span>
;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">EO2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/><a NAME="E9:48_1_1_2"/><span class="lab"><font color="Green" title="E51">A49</font></span>: 
<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="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="c6">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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E1:48_1_1_2"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_1_2_3"/><span class="lab"><font color="Green" title="E52">A50</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:48_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A51</font></span>: 
<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">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:48_1_1_2_3"/><span class="lab"><font color="Green" title="E54">A52</font></span>: 
<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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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> <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_2"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="txt" href="#E1:48_1_1_2_3"><span class="lab"><font color="Green" title="E52">A50</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>;<br/><a NAME="E4:48_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_3"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="ref" href="glib_000.html#D27" title="GLIB_000:def.27">GLIB_000:def 27</a></span>;<br/><a NAME="E5:48_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A53</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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="c6">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="c6">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="48_1_1_2_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_3_1_1_1"/><span class="lab"><font color="Green" title="E56">A54</font></span>: 
<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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><a NAME="E2:48_1_1_2_3_1_1_1"/><span class="kw">then </span>
<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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:48_1_1_2_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" title="FUNCOP_1:th.72">FUNCOP_1:72</a></span>
<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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_1_1_2"><span class="lab"><font color="Green" title="E51">A49</font></span></a>, <a class="txt" href="#E2:48_1_1_2_3"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="txt" href="#E1:48_1_1_2_3_1_1_1"><span class="lab"><font color="Green" title="E56">A54</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_1_1_2"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="txt" href="#E1:48_1_1_2_3"><span class="lab"><font color="Green" title="E52">A50</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_3_1_1_2"/><span class="lab"><font color="Green" title="E56">A55</font></span>: 
<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="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><div class="add"><div><span class="lab"><font color="Green" title="E57">A56</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_2_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A57</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A58</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:48_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A59</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:48_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A60</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_2_3_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A61</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_1_1_2_3_1_1_2_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_1_1_2_3"><span class="lab"><font color="Green" title="E55">A53</font></span></a>, <a class="txt" href="#E7:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E62">A61</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A62</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A63</font></span>: 
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E3:48_1_1_2_3_1_1_2_1_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="txt" href="#E1:48_1_1_2_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E4:48_1_1_2_3_1_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E64">A63</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_2_3_1_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E59">A58</font></span></a>, <a class="txt" href="#E1:48_1_1_2_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E63">A62</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_3_1_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A64</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E6:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E61">A60</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_3_1_1_2_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_1_1_2_3_1_1_2_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_2_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_3_1_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:48_1_1_2_3_1_1_2_1_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_3_1_1_2_1_1_1_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:48_1_1_2_3_1_1_2_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A57</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_2_3_1_1_2_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="txt" href="#E5:48_1_1_2_3_1_1_2_1"><span class="lab"><font color="Green" title="E60">A59</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:48_1_1_2_3_1_1_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:48_1_1_2_3_1_1_2"/>
 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="c6">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E56">A55</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E4:48_1_1_2_3_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="c6">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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c2">EL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_3"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E2:48_1_1_2_3"><span class="lab"><font color="Green" title="E53">A51</font></span></a>, <a class="txt" href="#E3:48_1_1_2_3"><span class="lab"><font color="Green" title="E54">A52</font></span></a>, <a class="txt" href="#E2:48_1_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E57">A56</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_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="c13">EO2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A65</font></span>:  <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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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#K2" title="XCMPLX_0:func.2">+</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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="funct_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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_1.html#K1" title="FUNCT_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="c6">lenP2g</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_1_1_2"><span class="lab"><font color="Green" title="E50">A48</font></span></a>, <a class="txt" href="#E6:48_1_1_2"><span class="lab"><font color="Green" title="E49">A47</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a>, <a class="ref" href="glib_004.html#T9" title="GLIB_004:th.9">GLIB_004:9</a></span>
<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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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="E12:48_1_1_2"/><span class="lab"><font color="Green" title="E53">A66</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><div><span class="lab"><font color="Green" title="E54">A67</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> )</span><br/><span class="kw">assume </span><a NAME="E1:48_1_1_2_5"/><span class="lab"><font color="Green" title="E55">A68</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font></span><br/><a NAME="E2:48_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A69</font></span>: 
<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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E3:48_1_1_2_5"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D26" title="GLIB_000:def.26">GLIB_000:def 26</a></span>;<br/><a NAME="E4:48_1_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A70</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:48_1_1_2_5_1"/><span class="lab"><font color="Green" title="E58">A71</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:48_1_1_2_5_1"/><span class="lab"><font color="Green" title="E59">A72</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:48_1_1_2_5_1"/><span class="lab"><font color="Green" title="E60">A73</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:48_1_1_2_5_1"/><span class="lab"><font color="Green" title="E61">A74</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:48_1_1_2_5_1"/><span class="lab"><font color="Green" title="E62">A75</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T103" title="GLIB_001:th.103">GLIB_001:103</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_5_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_5_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:48_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_1_1_2_5"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="txt" href="#E7:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E62">A75</font></span></a>, <a class="ref" href="glib_000.html#D13" title="GLIB_000:def.13">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_5_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_1_1_2_5_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A76</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_5_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A77</font></span>: 
<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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E58">A71</font></span></a>, <a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></span>;<br/><a NAME="E3:48_1_1_2_5_1_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E59">A72</font></span></a>, <a class="txt" href="#E1:48_1_1_2_5_1_1_1_1"><span class="lab"><font color="Green" title="E63">A76</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><a NAME="E4:48_1_1_2_5_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_5_1_1_1_1"><span class="lab"><font color="Green" title="E64">A77</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E5:48_1_1_2_5_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E4:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E59">A72</font></span></a>, <a class="txt" href="#E1:48_1_1_2_5_1_1_1_1"><span class="lab"><font color="Green" title="E63">A76</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_1_2_5_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_1_1_2_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A78</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E6:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E61">A74</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1_2_5_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:48_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:48_1_1_2_5_1_1_1_2_1"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_5_1_1_1_2"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><a NAME="E3:48_1_1_2_5_1_1_1_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_1_1_2_5_1_1_1_2_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:48_1_1_2_5_1_1_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E58">A71</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:48_1_1_2_5_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A79</font></span>: 
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">sink</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2"><span class="lab"><font color="Green" title="E47">A45</font></span></a>, <a class="txt" href="#E4:48_1_1_2_5"><span class="lab"><font color="Green" title="E57">A70</font></span></a>, <a class="txt" href="#E5:48_1_1_2_5_1"><span class="lab"><font color="Green" title="E60">A73</font></span></a>, <a class="ref" href="glib_000.html#D14" title="GLIB_000:def.14">GLIB_000:def 14</a></span>;<br/><a NAME="E6:48_1_1_2_5_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">lenP2g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="glib_001.html#D28" title="GLIB_001:def.28">GLIB_001:def 28</a></span>;<br/><span class="kw">hence </span><a NAME="E7:48_1_1_2_5_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E5:48_1_1_2_5_1_1_1_2"><span class="lab"><font color="Green" title="E64">A79</font></span></a>, <a class="ref" href="glib_001.html#T17" title="GLIB_001:th.17">GLIB_001:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E9:48_1_1_2_5_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">then </span><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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font> = 
<font color="Maroon" title="c2">EL</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E2:48_1_1_2_5"><span class="lab"><font color="Green" title="E56">A69</font></span></a>, <a class="ref" href="glib_005.html#D17" target="_self" title="GLIB_005:def.17">Def17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_1_1_2_5"><span class="lab"><font color="Green" title="E56">A69</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:48_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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_1_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="txt" href="#E1:48_1_1_2_5"><span class="lab"><font color="Green" title="E55">A68</font></span></a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E14:48_1_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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#K18" title="GLIB_000:func.18">.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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><span class="kw">hence </span><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="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48_1_1_2"><span class="lab"><font color="Green" title="E52">A65</font></span></a>, <a class="txt" href="#E12:48_1_1_2"><span class="lab"><font color="Green" title="E53">A66</font></span></a>, <a class="txt" href="#E13:48_1_1_2"><span class="lab"><font color="Green" title="E54">A67</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.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="relat_1.html#K7" title="RELAT_1:func.7">|</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">.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>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:48"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
