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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> is <font color="Maroon" title="c1">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  implies <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> ,<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <a href="glib_015.html#R1" title="GLIB_015:pred.1">are_Disomorphic</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:230"/>
<font color="Maroon" title="c2">G2</font> is <font color="Maroon" title="c1">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</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="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> ,<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <a href="glib_015.html#R1" title="GLIB_015:pred.1">are_Disomorphic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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><span class="kw"> such that </span><br/><a NAME="E3:230"/><span class="lab"><font color="Green" title="E140">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D24" onmouseover="rs('glib_010/D24')" onmouseout="rh()">GLIB_010:def 24</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">f</font> = <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span>;<br/>
<a NAME="E4:230"/><span class="lab"><font color="Green" title="E141">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:230"/><span class="lab"><font color="Green" title="E142">A3</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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:230"><span class="lab"><font color="Green" title="E140">A1</font></span></a>, <a class="ref" href="glenum00.html#T180" target="_self" onmouseover="rs('glenum00/T180')" onmouseout="rh()">Th180</a></span>;<br/>
<a NAME="E6:230"/>
 <a href="glenum00.html#K5" title="GLENUM00:func.5">SG2SGFunc</a> <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:230"><span class="lab"><font color="Green" title="E140">A1</font></span></a>, <a class="ref" href="glenum00.html#T31" target="_self" onmouseover="rs('glenum00/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E7:230"/><span class="kw">then </span><span class="lab"><font color="Green" title="E143">A4</font></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"><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="230_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">G</font> being   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  holds <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">G</font> is  <font color="Olive">b<sub>1</sub></font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">G</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="c5">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  implies <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">G</font> is  <font color="Maroon" title="c5">G</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> )</span><br/><span class="kw">assume </span><a NAME="E1:230_1"/><span class="lab"><font color="Green" title="E144">A5</font></span>: 
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</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"> <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">G</font> is  <font color="Maroon" title="c5">G</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c5">G</font> as   <a href="glib_000.html#V9" title="GLIB_000:attr.9">spanning</a>   <a href="glib_002.html#V2" title="GLIB_002:attr.2">acyclic</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="ref" href="glenum00.html#T168" target="_self" onmouseover="rs('glenum00/T168')" onmouseout="rh()">Th168</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">F9</font> = <font color="Maroon" title="c3">F</font> <a href="glib_010.html#K10" title="GLIB_010:func.10">|</a> <font color="Maroon" title="c6">H</font> as    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c6">H</font>, <a href="glib_010.html#K9" title="GLIB_010:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K10" title="GLIB_010:func.10">|</a> <font color="Maroon" title="c6">H</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre1.html#T88" onmouseover="rs('glibpre1/T88')" onmouseout="rh()">GLIBPRE1:88</a></span>;<br/><span class="lab"><font color="Green" title="E145">A6</font></span>: <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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">G</font> = 
<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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">G</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:230_1"><span class="lab"><font color="Green" title="E144">A5</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="glib_010.html#K9" title="GLIB_010:func.9">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K10" title="GLIB_010:func.10">|</a> <font color="Maroon" title="c6">H</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#D5" target="_self" onmouseover="rs('glenum00/D5')" onmouseout="rh()">Def5</a></span>
;<br/><a NAME="E5:230_1"/>
<font color="Maroon" title="c7">F9</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:230"><span class="lab"><font color="Green" title="E140">A1</font></span></a>, <a class="ref" href="glibpre1.html#T110" onmouseover="rs('glibpre1/T110')" onmouseout="rh()">GLIBPRE1:110</a></span>;<br/><span class="kw">hence </span><a NAME="E6:230_1"/>
<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#K14" title="GLENUM00:func.14">.allSpanningTrees()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">G</font> is  <font color="Maroon" title="c5">G</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:230_1"><span class="lab"><font color="Green" title="E145">A6</font></span></a>, <a class="ref" href="glib_010.html#D24" onmouseover="rs('glib_010/D24')" onmouseout="rh()">GLIB_010:def 24</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="E9:230"/>
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a> ,<font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K14" title="GLENUM00:func.14">.allSpanningTrees()</a>  <a href="glib_015.html#R1" title="GLIB_015:pred.1">are_Disomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:230"><span class="lab"><font color="Green" title="E141">A2</font></span></a>, <a class="txt" href="#E5:230"><span class="lab"><font color="Green" title="E142">A3</font></span></a>, <a class="txt" href="#E7:230"><span class="lab"><font color="Green" title="E143">A4</font></span></a>, <a class="ref" href="glib_015.html#D12" onmouseover="rs('glib_015/D12')" onmouseout="rh()">GLIB_015:def 12</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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