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

<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W1</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Olive" title="b1">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W2</font> holds <br/>( ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) &amp; ( <font color="Olive" title="b2">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Olive" title="b1">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">W1</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">W2</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">W2</font> holds <br/>( ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) &amp; ( <font color="Olive" title="b1">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) )</span><br/><b>let </b><font color="Maroon" title="c4">W2</font> be    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c2">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="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> implies ( ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) ) )</span><br/>







<b>assume </b><a NAME="E1:252"/><i><font color="Green" title="E154">A1</font></i>: 
( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) )</span><br/>

<a NAME="E2:252"/><b>then </b>
<font color="Maroon" title="c1">G1</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c2">G2</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T90" title="GLIB_000:th.90">GLIB_000:90</a></i>;<br/>
<b>hence </b><a NAME="E3:252"/>
( ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V1" title="GLIB_001:attr.1">closed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V2" title="GLIB_001:attr.2">directed</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V4" title="GLIB_001:attr.4">Trail-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> ) &amp; ( <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) &amp; ( <font color="Maroon" title="c4">W2</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> implies <font color="Maroon" title="c3">W1</font> is <a href="glib_001.html#V6" title="GLIB_001:attr.6">vertex-distinct</a> ) )
 <b>by </b><i><a class="txt" href="glib_001.html#E1:252"><i><font color="Green" title="E154">A1</font></i></a>, <a class="ref" href="glib_001.html#T177" target="_self" title="GLIB_001:th.177">Th177</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>
