<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="net_1.html#NM1" title="NET_1:NM.1">Pnet</a>; <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="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">T</font> =  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:18"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:18_1"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/>



<span class="kw">assume </span><a NAME="E1:18_1_1"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"> contradiction</span><br/>

<a NAME="E2:18_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:18_1_1"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_1"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:18_1_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E5:18_1_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:18_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_1_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/>
<a NAME="E7:18_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_1_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="ff_siec.html#T7" target="_self" title="FF_SIEC:th.7">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:18_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_1_1"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</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="E2:18_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T37" title="RELAT_1:th.37">RELAT_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:18"/>
<span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:18_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/>



<span class="kw">assume </span><a NAME="E1:18_2_1"/><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"> contradiction</span><br/>

<a NAME="E2:18_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:18_2_1"/><span class="lab"><font color="Green" title="E12">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_2_1"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:18_2_1"/><span class="lab"><font color="Green" title="E13">A9</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E5:18_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_2_1"><span class="lab"><font color="Green" title="E11">A7</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:18_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_2_1"><span class="lab"><font color="Green" title="E13">A9</font></span></a>, <a class="ref" href="ff_siec.html#T7" target="_self" title="FF_SIEC:th.7">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_2_1"><span class="lab"><font color="Green" title="E12">A8</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</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="E2:18_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T37" title="RELAT_1:th.37">RELAT_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_2"/>
<span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E3:18"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:18_3"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/>



<span class="kw">assume </span><a NAME="E1:18_3_1"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"> contradiction</span><br/>

<a NAME="E2:18_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A11</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:18_3_1"/><span class="lab"><font color="Green" title="E12">A12</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_3_1"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:18_3_1"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E5:18_3_1"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_3_1"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:18_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_3_1"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="relat_1.html#D7" title="RELAT_1:def.7">RELAT_1:def 7</a></span>;<br/>
<a NAME="E7:18_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18_3_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="ff_siec.html#T7" target="_self" title="FF_SIEC:th.7">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:18_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_3_1"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</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="E2:18_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T37" title="RELAT_1:th.37">RELAT_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:18_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K4" title="RELAT_1:func.4">~</a></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">set </span><font color="Maroon" title="c3">R</font> = <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E4:18"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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">  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span><br/>



<span class="kw">assume </span><a NAME="E1:18_4"/><span class="lab"><font color="Green" title="E10">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></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"> contradiction</span><br/>

<a NAME="E2:18_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:18_4"/><span class="lab"><font color="Green" title="E12">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_4"><span class="lab"><font color="Green" title="E10">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E4:18_4"/><span class="lab"><font color="Green" title="E13">A18</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_4"><span class="lab"><font color="Green" title="E11">A16</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E5:18_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_4"><span class="lab"><font color="Green" title="E11">A16</font></span></a>, <a class="ref" href="relat_1.html#D11" title="RELAT_1:def.11">RELAT_1:def 11</a></span>;<br/>
<a NAME="E6:18_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18_4"><span class="lab"><font color="Green" title="E13">A18</font></span></a>, <a class="ref" href="ff_siec.html#T7" target="_self" title="FF_SIEC:th.7">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:18_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18_4"><span class="lab"><font color="Green" title="E12">A17</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</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="E5:18"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p2">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T37" title="RELAT_1:th.37">RELAT_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:18"/>
<span class="p1">(<span class="default"><a href="net_1.html#K1" title="NET_1:func.1">Flow</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">M</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="relat_1.html#K6" title="RELAT_1:func.6">id</a> <span class="p1">(<span class="default"><a href="net_1.html#K2" title="NET_1:func.2">Elements</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</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>
