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

<b>let </b><font color="Maroon" title="c3">G1</font>, <font color="Maroon" title="c4">G2</font> be   <a href="graph_1.html#NM1" title="GRAPH_1:NM.1">Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> implies <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E39">A1</font></i>: 
( the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U3" title="GRAPH_1:sel.3">Source</a> of <font color="Maroon" title="c4">G2</font> &amp; the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c3">G1</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> the <a href="graph_1.html#U4" title="GRAPH_1:sel.4">Target</a> of <font color="Maroon" title="c4">G2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span><br/>

<a NAME="E2:69"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font> implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">v</font> as  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c3">G1</font> <b>by </b><i><a class="ref" href="graph_1.html#D24" target="_self" title="GRAPH_1:def.24">Def24</a></i>;<br/>
<a NAME="E3:69_1"/>
( <font color="Maroon" title="c6">G</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c3">G1</font> &amp; <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:69"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a>, <a class="ref" href="graph_1.html#T19" target="_self" title="GRAPH_1:th.19">Th19</a></i>;<br/>
<a NAME="E4:69_1"/><b>then </b>
<font color="Maroon" title="c6">G</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#T17" target="_self" title="GRAPH_1:th.17">Th17</a></i>;<br/>
<a NAME="E5:69_1"/><b>then </b>
<font color="Maroon" title="c6">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a></i>;<br/>
<b>hence </b><a NAME="E6:69_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_1.html#D24" target="_self" title="GRAPH_1:def.24">Def24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:69"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font> implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:69_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">G</font> = <font color="Maroon" title="c5">v</font> as  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c4">G2</font> <b>by </b><i><a class="ref" href="graph_1.html#D24" target="_self" title="GRAPH_1:def.24">Def24</a></i>;<br/>
<a NAME="E3:69_2"/>
( <font color="Maroon" title="c6">G</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c4">G2</font> &amp; <font color="Maroon" title="c4">G2</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font> )
 
<b>by </b><i><a class="txt" href="graph_1.html#E1:69"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a>, <a class="ref" href="graph_1.html#T19" target="_self" title="GRAPH_1:th.19">Th19</a></i>;<br/>
<a NAME="E4:69_2"/><b>then </b>
<font color="Maroon" title="c6">G</font> <a href="graph_1.html#R4" title="GRAPH_1:pred.4">c=</a> <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#T17" target="_self" title="GRAPH_1:th.17">Th17</a></i>;<br/>
<a NAME="E5:69_2"/><b>then </b>
<font color="Maroon" title="c6">G</font> is  <a href="graph_1.html#V1" title="GRAPH_1:attr.1">strict</a>  <a href="graph_1.html#M3" title="GRAPH_1:mode.3">Subgraph</a> of <font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font>
 
<b>by </b><i><a class="ref" href="graph_1.html#D23" target="_self" title="GRAPH_1:def.23">Def23</a></i>;<br/>
<b>hence </b><a NAME="E6:69_2"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="graph_1.html#D24" target="_self" title="GRAPH_1:def.24">Def24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:69"/>
for <font color="Olive" title="b1">v</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span> holds <br/><font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span> implies <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:69_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span></span><br/>

<a NAME="E2:69_3"/><b>then </b>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font> or <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:69_3"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="txt" href="graph_1.html#E2:69"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="graph_1.html#E3:69"><i><font color="Green" title="E41">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:69"/>
<span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c3">G1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <font color="Maroon" title="c4">G2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_1.html#K7" title="GRAPH_1:func.7">bool</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G1</font> <a href="graph_1.html#K1" title="GRAPH_1:func.1">\/</a> <font color="Maroon" title="c4">G2</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
