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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="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  iff <font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font> )</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="c2">G2</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:98_1"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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">G2</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E2:98_1"/>
<font color="Maroon" title="c2">G2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T62" target="_self" onmouseover="rs('glenum00/T62')" onmouseout="rh()">Th62</a></span>;<br/><a NAME="E3:98_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">G2</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">|</a> <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <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="txt" href="#E1:98_1"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</a></span>;<br/><span class="kw">hence </span><a NAME="E4:98_1"/>
<font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <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="glib_000.html#T171" onmouseover="rs('glib_000/T171')" onmouseout="rh()">GLIB_000:171</a>, <a class="ref" href="glib_000.html#T128" onmouseover="rs('glib_000/T128')" onmouseout="rh()">GLIB_000:128</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="E2:98"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c2">G2</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c1">G1</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="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="98_2"><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>   st <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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</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="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:98_2"/>
<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#K8" title="GLENUM00:func.8">.allSpanningSG()</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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> </span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">H</font> = <font color="Maroon" title="c3">x</font> as   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</a></span>;<br/> <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">H</font> = 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>
<br/>.= 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:98"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>
;<br/><a NAME="E4:98_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">H</font> is   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>   <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="txt" href="#E2:98"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="ref" href="glib_000.html#T43" onmouseover="rs('glib_000/T43')" onmouseout="rh()">GLIB_000:43</a>, <a class="ref" href="glib_000.html#D33" onmouseover="rs('glib_000/D33')" onmouseout="rh()">GLIB_000:def 33</a></span>;<br/><span class="kw">hence </span><a NAME="E5:98_2"/>
<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#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T60" target="_self" onmouseover="rs('glenum00/T60')" onmouseout="rh()">Th60</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="E4:98"/>
<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K8" title="GLENUM00:func.8">.allSpanningSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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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