<?xml version="1.0"?>
<div class="add">

<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#V7" title="GLIB_010:attr.7">total</a>  &amp; <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#V7" title="GLIB_010:attr.7">total</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
( <font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span>;<br/>
<a NAME="E2:73"/><span class="lab"><font color="Green" title="E48">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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="73_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>  <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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span></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="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span></span>)</span> implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:73_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span></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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:73_1"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span></span>)</span> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K7" title="GLENUM00:func.7">.allInducedSG()</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">V1</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:73_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c7">V1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E3:73_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">H1</font> = <font color="Maroon" title="c6">y</font> as   <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c7">V1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a></span>;<br/><a NAME="E7:73_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="#E1:73"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="glib_010.html#D11" onmouseover="rs('glib_010/D11')" onmouseout="rh()">GLIB_010:def 11</a></span>;<br/><a NAME="E8:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V1</font> is   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T119" onmouseover="rs('relat_1/T119')" onmouseout="rh()">RELAT_1:119</a></span>;<br/><a NAME="E9:73_1"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a>  <a href="glib_010.html#K8" title="GLIB_010:func.8">dom</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="glib_010.html#T55" onmouseover="rs('glib_010/T55')" onmouseout="rh()">GLIB_010:55</a></span>;<br/><a NAME="E10:73_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A6</font></span>: 
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <span class="p1">(<span class="default"><a href="glib_010.html#K8" title="GLIB_010:func.8">dom</a> <font color="Maroon" title="c3">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D34" onmouseover="rs('glib_000/D34')" onmouseout="rh()">GLIB_000:def 34</a></span>;<br/><font color="Maroon" title="c5">x</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="c6">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:73_1"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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="c8">H1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:73_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="glenum00.html#D5" target="_self" onmouseover="rs('glenum00/D5')" onmouseout="rh()">Def5</a></span>
;<br/><a NAME="E12:73_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> is   <a href="glib_009.html#V1" title="GLIB_009:attr.1">plain</a>  <a href="glib_000.html#NM5" title="GLIB_000:NM.5">inducedSubgraph</a> of <font color="Maroon" title="c2">G2</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c7">V1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E10:73_1"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="ref" href="glibpre1.html#T102" onmouseover="rs('glibpre1/T102')" onmouseout="rh()">GLIBPRE1:102</a></span>;<br/><span class="kw">hence </span><a NAME="E13:73_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glenum00.html#K7" title="GLENUM00:func.7">.allInducedSG()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:73_1"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="ref" href="glenum00.html#T45" target="_self" onmouseover="rs('glenum00/T45')" onmouseout="rh()">Th45</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="E4:73"/>
 <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#K7" title="GLENUM00:func.7">.allInducedSG()</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#K7" title="GLENUM00:func.7">.allInducedSG()</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/>


</div>
