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

<span class="kw">set </span><font color="Maroon" title="c1">V</font> = <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">E</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E1:8_1_1"/>
( 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; 2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">S</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 1, <font color="Maroon" title="c4">T</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> 2 as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T45" onmouseover="rs('funcop_1/T45')" onmouseout="rh()">FUNCOP_1:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">G</font> =  <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(# <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>,<font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">T</font> #) as   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">G</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  &amp;  not <font color="Maroon" title="c5">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:8_1_1"/>
<font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V6" title="GRAPH_1:attr.6">finite</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  &amp;  not <font color="Maroon" title="c5">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  )</span><br/>

<a NAME="E5:8_1_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:8_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:8_1_1"/>
<font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V4" title="GRAPH_1:attr.4">simple</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a>  &amp;  not <font color="Maroon" title="c5">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><a NAME="E2:8_1_1_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">G</font>
 <span class="kw">and </span><br/><a NAME="E3:8_1_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="graph_1.html#D9" onmouseover="rs('graph_1/D9')" onmouseout="rh()">GRAPH_1:def 9</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:8_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:8_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E3:8_1_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E8:8_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:8_1_1"/>
<font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V5" title="GRAPH_1:attr.5">connected</a> 
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  &amp; <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="8_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c6">MSG</font> =  <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G</font> #);<br/>
<span class="kw">given </span><font color="Maroon" title="c7">G1</font>, <font color="Maroon" title="c8">G2</font> being   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">Graph</a><span class="kw"> such that </span><a NAME="E2:8_1_1_2"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">and </span><br/><a NAME="E3:8_1_1_2"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="graph_1.html#R1" title="GRAPH_1:pred.1">is_sum_of</a> <font color="Maroon" title="c7">G1</font>,<font color="Maroon" title="c8">G2</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="graph_1.html#D10" onmouseover="rs('graph_1/D10')" onmouseout="rh()">GRAPH_1:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:8_1_1_2"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
 <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G1</font> <a href="graph_1.html#K5" title="GRAPH_1:func.5">\/</a> <font color="Maroon" title="c8">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">V1</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">V2</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">T1</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">T2</font> =  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">S1</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">S2</font> =  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">E1</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">E2</font> =  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font>;<br/>
<a NAME="E5:8_1_1_2"/><span class="lab"><font color="Green" title="E9">A9</font></span>: 
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#T3" target="_self" onmouseover="rs('msscyc_1/T3')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E6:8_1_1_2"/><span class="lab"><font color="Green" title="E10">A10</font></span>: 
<span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msscyc_1.html#T3" target="_self" onmouseover="rs('msscyc_1/T3')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E7:8_1_1_2"/><span class="lab"><font color="Green" title="E11">A11</font></span>: 
(  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font> &amp;  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:8_1_1_2"><span class="lab"><font color="Green" title="E7">A7</font></span></a></span>;<br/>
<a NAME="E8:8_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A12</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/>
<a NAME="E9:8_1_1_2"/><span class="lab"><font color="Green" title="E13">A13</font></span>: 
 the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <a href="graph_1.html#G1" title="GRAPH_1:aggr.1">MultiGraphStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>, the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c5">G</font>, the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c5">G</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:8_1_1_2_1"/>
( (  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or (  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) or (  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:8_1_1_2"><span class="lab"><font color="Green" title="E13">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" onmouseover="rs('zfmisc_1/T37')" onmouseout="rh()">ZFMISC_1:37</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2_1_1"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
(  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:8_1_1_2_1_1"/><span class="kw">then </span>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E3:8_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E4:8_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A15</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:8_1_1_2_1_1"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E6:8_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_1"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E7:8_1_1_2_1_1"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:8_1_1_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1_2"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E5:8_1_1_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E6:8_1_1_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E4:8_1_1_2_1_1"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="8_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2_1_2"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
(  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:8_1_1_2_1_2"/><span class="kw">then </span>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E3:8_1_1_2_1_2"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E4:8_1_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A17</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:8_1_1_2_1_2"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E6:8_1_1_2_1_2"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E7:8_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_2"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E8:8_1_1_2_1_2"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c7">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c7">G1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:8_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1_2"><span class="lab"><font color="Green" title="E9">A9</font></span></a>, <a class="txt" href="#E4:8_1_1_2_1_2"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E10:8_1_1_2_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1_2_1_2"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/><a NAME="E11:8_1_1_2_1_2"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:8_1_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1_2"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="xboole_1.html#T67" onmouseover="rs('xboole_1/T67')" onmouseout="rh()">XBOOLE_1:67</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="8_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:8_1_1_2_1_3"/><span class="lab"><font color="Green" title="E14">A19</font></span>: 
(  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c7">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c8">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E2:8_1_1_2_1_3"/><span class="kw">then </span>
 the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E3:8_1_1_2_1_3"/><span class="kw">then </span>
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E8:8_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E4:8_1_1_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A20</font></span>: 
2 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E5:8_1_1_2_1_3"/><span class="lab"><font color="Green" title="E16">A21</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><a NAME="E6:8_1_1_2_1_3"/>
 the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E7:8_1_1_2"><span class="lab"><font color="Green" title="E11">A11</font></span></a>, <a class="txt" href="#E4:8_1_1_2"><span class="lab"><font color="Green" title="E8">A8</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="ref" href="graph_1.html#D5" onmouseover="rs('graph_1/D5')" onmouseout="rh()">GRAPH_1:def 5</a></span>;<br/><a NAME="E7:8_1_1_2_1_3"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E6:8_1_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:8_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A19</font></span></a>, <a class="ref" href="funct_2.html#T4" onmouseover="rs('funct_2/T4')" onmouseout="rh()">FUNCT_2:4</a></span>;<br/><a NAME="E8:8_1_1_2_1_3"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c8">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a>  the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c8">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E9:8_1_1_2_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:8_1_1_2"><span class="lab"><font color="Green" title="E10">A10</font></span></a>, <a class="txt" href="#E4:8_1_1_2_1_3"><span class="lab"><font color="Green" title="E15">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" onmouseover="rs('zfmisc_1/T32')" onmouseout="rh()">ZFMISC_1:32</a></span>;<br/><a NAME="E10:8_1_1_2_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:8_1_1_2_1_3"><span class="lab"><font color="Green" title="E16">A21</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/><a NAME="E11:8_1_1_2_1_3"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:8_1_1_2"><span class="lab"><font color="Green" title="E12">A12</font></span></a>, <a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:8_1_1_2_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:8_1_1_2"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="ref" href="xboole_1.html#T67" onmouseover="rs('xboole_1/T67')" onmouseout="rh()">XBOOLE_1:67</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">thus </span><a NAME="E10:8_1_1"/>
 not <font color="Maroon" title="c5">G</font> is  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a> </span><br/>

<span class="kw">thus </span><a NAME="E11:8_1_1"/>
<font color="Maroon" title="c5">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
