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<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="214_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">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> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">H</font> where <font color="Olive" title="b2">H</font> is    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> : ( <font color="Olive" title="b2">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Olive" title="b2">H</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  ) </span> } </span>  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">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">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="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> : ( <font color="Olive" title="b1">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  ) </span> } </span>  implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:214_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> : ( <font color="Olive" title="b1">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  ) </span> } </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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">H</font> being    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:214_1_1_1"/><span class="lab"><font color="Green" title="E128">A1</font></span>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font>
 <span class="kw">and </span><br/><a NAME="E4:214_1_1_1"/>
( <font color="Maroon" title="c3">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Maroon" title="c3">H</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  )
 ;<br/><span class="kw">thus </span><a NAME="E5:214_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:214_1_1_1"><span class="lab"><font color="Green" title="E128">A1</font></span></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="E2:214_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">H</font> where <font color="Olive" title="b1">H</font> is    <a href="glenum00.html#M1" title="GLENUM00:mode.1">Element</a> of  <a href="subset_1.html#K2" title="SUBSET_1:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span> : ( <font color="Olive" title="b1">H</font> is  <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>  &amp; <font color="Olive" title="b1">H</font> is  <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>  ) </span> } </span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glenum00.html#K1" title="GLENUM00:func.1">.allSG()</a></span>)</span>
 <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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