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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">  for <font color="Olive" title="b1">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  holds <br/> <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<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="c3">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  <a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font>;<br/>
<span class="p1">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span> = 
<span class="p1">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T139" target="_self" onmouseover="rs('glenum00/T139')" onmouseout="rh()">Th139</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T79" onmouseover="rs('relat_1/T79')" onmouseout="rh()">RELAT_1:79</a></span>
;<br/>
<a NAME="E2:192"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T13" onmouseover="rs('relat_1/T13')" onmouseout="rh()">RELAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:192"/>
<font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</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">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> </span><br/>

<a NAME="E4:192"/><span class="kw">then </span>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a>  &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T90" target="_self" onmouseover="rs('glenum00/T90')" onmouseout="rh()">Th90</a>, <a class="ref" href="glenum00.html#T132" target="_self" onmouseover="rs('glenum00/T132')" onmouseout="rh()">Th132</a></span>;<br/>
<a NAME="E5:192"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T27" onmouseover="rs('xboole_1/T27')" onmouseout="rh()">XBOOLE_1:27</a></span>;<br/>
<a NAME="E6:192"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K11" title="GLENUM00:func.11">.allConnectedSG()</a></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K9" title="GLENUM00:func.9">.allForests()</a></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T139" target="_self" onmouseover="rs('glenum00/T139')" onmouseout="rh()">Th139</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:192"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="funct_3.html#K17" title="FUNCT_3:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:192"><span class="lab"><font color="Green" title="E115">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</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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