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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">X</font>, <font color="Olive" title="b5">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> holds <br/>( ( <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Olive" title="b1">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Olive" title="b1">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Olive" title="b4">X</font>,<font color="Olive" title="b5">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Olive" title="b1">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Olive" title="b4">X</font>,<font color="Olive" title="b5">Y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Olive" title="b1">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Olive" title="b4">X</font>,<font color="Olive" title="b5">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Olive" title="b1">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Olive" title="b4">X</font>,<font color="Olive" title="b5">Y</font>,<font color="Maroon" title="c2">G2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">e</font>, <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">X</font>, <font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> implies ( ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) ) )</span><br/>













<span class="kw">assume </span><a NAME="E1:184"/>
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) )</span><br/>

<a NAME="E2:184"/><span class="kw">then </span>
<font color="Maroon" title="c1">G1</font> is    <a href="glib_000.html#M1" title="GLIB_000:mode.1">Subgraph</a> of <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T90" target="_self" title="GLIB_000:th.90">Th90</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:184"/>
( ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R3" title="GLIB_000:pred.3">SJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) &amp; ( <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c1">G1</font> implies <font color="Maroon" title="c3">e</font> <a href="glib_000.html#R4" title="GLIB_000:pred.4">DSJoins</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c7">Y</font>,<font color="Maroon" title="c2">G2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T75" target="_self" title="GLIB_000:th.75">Th75</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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