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<span class="kw">let </span><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" 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">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> iff <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <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> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_1"/><span class="lab"><font color="Green" title="E147">A1</font></span>: 
<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>
 ; <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">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a> </span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="246_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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  ) )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">x</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  ) &amp; ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  implies <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:246_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">x</font> as   <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">G1</font> <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/><a NAME="E3:246_1_1_1"/>
<font color="Maroon" title="c4">C</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:246_1"><span class="lab"><font color="Green" title="E147">A1</font></span></a>, <a class="ref" href="glibpre1.html#T39" onmouseover="rs('glibpre1/T39')" onmouseout="rh()">GLIBPRE1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E4:246_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</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>; <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="E2:246_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">x</font> as   <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="c2">G2</font> <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/><a NAME="E4:246_1_1"/>
<font color="Maroon" title="c4">C</font> is   <a href="glib_002.html#NM1" title="GLIB_002:NM.1">Component</a> of <font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:246_1"><span class="lab"><font color="Green" title="E147">A1</font></span></a>, <a class="ref" href="glibpre1.html#T39" onmouseover="rs('glibpre1/T39')" onmouseout="rh()">GLIBPRE1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:246_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G1</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>; <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="E3:246_1"/>
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:246"/>
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K15" title="GLENUM00:func.15">.allComponents()</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">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font></span><br/>

<a NAME="E3:246"/><span class="kw">then </span>
( <font color="Maroon" title="c1">G1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c2">G2</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T193" target="_self" onmouseover="rs('glenum00/T193')" onmouseout="rh()">Th193</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:246"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T87" onmouseover="rs('glib_000/T87')" onmouseout="rh()">GLIB_000:87</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/>


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