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<a NAME="E1:244_1_1"/>
 the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T189" target="_self" onmouseover="rs('glenum00/T189')" onmouseout="rh()">Th189</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:244_1_1"/>
 not <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_015.html#V1" title="GLIB_015:attr.1">vertex-disjoint</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_015.html#V2" title="GLIB_015:attr.2">edge-disjoint</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V14" title="GLIB_014:attr.14">\/-tolerating</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V2" title="GLIB_014:attr.2">plain</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V9" title="GLIB_014:attr.9">connected</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="244_1_1_1"><span class="kw">now </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">  for <font color="Olive" title="b1">G1</font>, <font color="Olive" title="b2">G2</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">G1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Olive" title="b2">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Olive" title="b1">G1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">G2</font> holds <br/> <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b1">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Olive" title="b2">G2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">G1</font>, <font color="Maroon" title="c3">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"> ( <font color="Maroon" title="c2">G1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Maroon" title="c2">G1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">G2</font> implies  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )</span><br/><span class="kw">set </span><font color="Maroon" title="c4">V1</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">V2</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>;<br/><span class="kw">assume </span><a NAME="E1:244_1_1_1"/><span class="lab"><font color="Green" title="E146">A1</font></span>: 
( <font color="Maroon" title="c2">G1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  &amp; <font color="Maroon" title="c2">G1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">G2</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">  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font></span><br/><a NAME="E2:244_1_1_1"/><span class="kw">then </span>
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a></span>)</span> &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_014.html#K1" title="GLIB_014:func.1">the_Vertices_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D14" onmouseover="rs('glib_014/D14')" onmouseout="rh()">GLIB_014:def 14</a></span>;<br/><a NAME="E3:244_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E147">A2</font></span>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  &amp;  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glenum00.html#E210"><span class="lab"><font color="Green" title="E145">Lm3</font></span></a></span>;<br/><a NAME="E4:244_1_1_1"/><span class="lab"><font color="Green" title="E148">A3</font></span>: 
<font color="Maroon" title="c1">G</font> <a href="glib_002.html#K3" title="GLIB_002:func.3">.componentSet()</a>  is    <a href="eqrel_1.html#M1" title="EQREL_1:mode.1">a_partition</a> of  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_008.html#T23" onmouseover="rs('glib_008/T23')" onmouseout="rh()">GLIB_008:23</a></span>;<br/><a NAME="E5:244_1_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="244_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:244_1_1_1_1"/><span class="lab"><font color="Green" title="E149">A4</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</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"> contradiction</span><br/>

<a NAME="E2:244_1_1_1_1"/>
( <font color="Maroon" title="c2">G1</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c3">G2</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:244_1_1_1"><span class="lab"><font color="Green" title="E146">A1</font></span></a>, <a class="ref" href="glenum00.html#T189" target="_self" onmouseover="rs('glenum00/T189')" onmouseout="rh()">Th189</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:244_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:244_1_1_1"><span class="lab"><font color="Green" title="E146">A1</font></span></a>, <a class="txt" href="#E1:244_1_1_1_1"><span class="lab"><font color="Green" title="E149">A4</font></span></a>, <a class="ref" href="glib_002.html#T32" onmouseover="rs('glib_002/T32')" onmouseout="rh()">GLIB_002:32</a>, <a class="ref" href="glib_009.html#T44" onmouseover="rs('glib_009/T44')" onmouseout="rh()">GLIB_009:44</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="E6:244_1_1_1"/>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:244_1_1_1"><span class="lab"><font color="Green" title="E147">A2</font></span></a>, <a class="txt" href="#E4:244_1_1_1"><span class="lab"><font color="Green" title="E148">A3</font></span></a>, <a class="ref" href="eqrel_1.html#D4" onmouseover="rs('eqrel_1/D4')" onmouseout="rh()">EQREL_1:def 4</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="E4:244_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_015.html#V1" title="GLIB_015:attr.1">vertex-disjoint</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_015.html#D18" onmouseover="rs('glib_015/D18')" onmouseout="rh()">GLIB_015:def 18</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="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_015.html#V2" title="GLIB_015:attr.2">edge-disjoint</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V14" title="GLIB_014:attr.14">\/-tolerating</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V2" title="GLIB_014:attr.2">plain</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V9" title="GLIB_014:attr.9">connected</a>  )</span><br/>

<span class="kw">hence </span><a NAME="E5:244_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_015.html#V2" title="GLIB_015:attr.2">edge-disjoint</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="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V14" title="GLIB_014:attr.14">\/-tolerating</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V2" title="GLIB_014:attr.2">plain</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V9" title="GLIB_014:attr.9">connected</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:244_1_1"/>
( <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V14" title="GLIB_014:attr.14">\/-tolerating</a>  &amp; <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V2" title="GLIB_014:attr.2">plain</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="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V9" title="GLIB_014:attr.9">connected</a> </span><br/>

<a NAME="E7:244_1_1"/>
 for <font color="Olive" title="b1">H</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">H</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  holds <br/><font color="Olive" title="b1">H</font> is  <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T189" target="_self" onmouseover="rs('glenum00/T189')" onmouseout="rh()">Th189</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:244_1_1"/>
<font color="Maroon" title="c1">G</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  is  <a href="glib_014.html#V9" title="GLIB_014:attr.9">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_014.html#D9" onmouseover="rs('glib_014/D9')" onmouseout="rh()">GLIB_014: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>
