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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</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">  for <font color="Olive" title="b1">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">E</font> holds <br/> ( <font color="Olive" title="b2">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Olive" title="b2">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">  for <font color="Olive" title="b1">G1</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</font> holds <br/> ( <font color="Olive" title="b1">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Olive" title="b1">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">E</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="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp; <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  )</span><br/>





<a NAME="E1:26"/>
 for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  iff <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">V</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  iff <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )</span><br/>

<div><span class="kw">hereby </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"> ( <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  implies <font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:26_1_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</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="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font><span class="kw"> such that </span><br/><a NAME="E3:26_1_1"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T10" target="_self" onmouseover="rs('glib_007/T10')" onmouseout="rh()">Th10</a></span>;<br/><a NAME="E5:26_1_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_1_1"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="ref" href="glib_007.html#T20" target="_self" onmouseover="rs('glib_007/T20')" onmouseout="rh()">Th20</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_1_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</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>

<span class="kw">assume </span><a NAME="E2:26_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</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="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G2</font><span class="kw"> such that </span><br/><a NAME="E4:26_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">w</font> = <font color="Maroon" title="c5">v</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T10" target="_self" onmouseover="rs('glib_007/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E6:26_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K1" title="GLIB_002:func.1">.reachableFrom</a> <font color="Maroon" title="c6">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="glib_007.html#T20" target="_self" onmouseover="rs('glib_007/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:26_1"/>
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D8" onmouseover="rs('glib_002/D8')" onmouseout="rh()">GLIB_002:def 8</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>
<span class="kw">hence </span><a NAME="E2:26"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</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"> <font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> </span><br/>

<span class="kw">thus </span><font color="Maroon" title="c3">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  = 
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="glib_002.html#D9" onmouseover="rs('glib_002/D9')" onmouseout="rh()">GLIB_002:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_002.html#D9" onmouseover="rs('glib_002/D9')" onmouseout="rh()">GLIB_002:def 9</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>
