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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">G3</font> being   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">G4</font> being   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <font color="Maroon" title="c2">G2</font> holds <br/> ( <font color="Olive" title="b1">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b2">G4</font> iff <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">G3</font> be   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</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">  for <font color="Olive" title="b1">G4</font> being   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <font color="Maroon" title="c2">G2</font> holds <br/> ( <font color="Maroon" title="c3">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Olive" title="b1">G4</font> iff <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c4">G4</font> be   <a href="glib_009.html#NM1" title="GLIB_009:NM.1">removeLoops</a> of <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">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</font> iff <font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</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#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  implies <font color="Maroon" title="c3">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:191_1"/><span class="lab"><font color="Green" title="E114">A1</font></span>: 
<font color="Maroon" title="c3">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</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#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> </span><br/><span class="kw">thus </span><font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  = 
<font color="Maroon" title="c3">G3</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#T146" target="_self" onmouseover="rs('glenum00/T146')" onmouseout="rh()">Th146</a></span>
<br/>.= 
<font color="Maroon" title="c4">G4</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:191_1"><span class="lab"><font color="Green" title="E114">A1</font></span></a>, <a class="ref" href="glenum00.html#T147" target="_self" onmouseover="rs('glenum00/T147')" onmouseout="rh()">Th147</a></span>
<br/>.= 
<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#T146" target="_self" onmouseover="rs('glenum00/T146')" onmouseout="rh()">Th146</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="E2:191"/><span class="lab"><font color="Green" title="E114">A2</font></span>: 
<font color="Maroon" title="c1">G1</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</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">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</font></span><br/>

<font color="Maroon" title="c3">G3</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a>  = 
<font color="Maroon" title="c1">G1</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#T146" target="_self" onmouseover="rs('glenum00/T146')" onmouseout="rh()">Th146</a></span>
<br/>.= 
<font color="Maroon" title="c4">G4</font> <a href="glenum00.html#K12" title="GLENUM00:func.12">.allTrees()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:191"><span class="lab"><font color="Green" title="E114">A2</font></span></a>, <a class="ref" href="glenum00.html#T146" target="_self" onmouseover="rs('glenum00/T146')" onmouseout="rh()">Th146</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:191"/>
<font color="Maroon" title="c3">G3</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glenum00.html#T147" target="_self" onmouseover="rs('glenum00/T147')" onmouseout="rh()">Th147</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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