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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">F0</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">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> ex <font color="Olive" title="b2">E</font> being   <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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b3">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b2">E</font>  ex <font color="Olive" title="b4">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="Olive" title="b3">G3</font> st <br/>( <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F0</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">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  ex <font color="Olive" title="b1">E</font> being   <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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">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="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:111"/>
<font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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">  ex <font color="Olive" title="b1">E</font> being   <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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">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="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">E</font> being   <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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:111"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
 for <font color="Olive" title="b1">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">E</font>  ex <font color="Olive" title="b2">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="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T86" target="_self" onmouseover="rs('glibpre0/T86')" onmouseout="rh()">Th92</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">E</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">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">E</font>  ex <font color="Olive" title="b2">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="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">G3</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">E</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">  ex <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="c5">G3</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">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="c5">G3</font><span class="kw"> such that </span><br/><a NAME="E5:111"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  )
 <span class="kw">and </span><br/><a NAME="E6:111"/>
(  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E7:111"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  )
 <span class="kw">and </span><br/><a NAME="E8:111"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  )
 <span class="kw">and </span><br/><a NAME="E9:111"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  )
 <span class="kw">and </span><br/><a NAME="E10:111"/>
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  )
 <span class="kw">and </span><br/><a NAME="E11:111"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">F</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="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:111"/>
( <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111"><span class="lab"><font color="Green" title="E31">A2</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"> ( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E13:111"/>
( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E11:111"><span class="lab"><font color="Green" title="E35">A6</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/>


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