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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="graph_1.html#NM3" title="GRAPH_1:NM.3">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">v1</font>, <font color="Olive" title="b2">v2</font>, <font color="Olive" title="b3">v3</font>, <font color="Olive" title="b4">v4</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b5">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font> &amp; <font color="Olive" title="b5">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Olive" title="b3">v3</font>,<font color="Olive" title="b4">v4</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v3</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v4</font> ) holds <br/>( <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v4</font> &amp; <font color="Olive" title="b2">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v3</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v1</font>, <font color="Maroon" title="c3">v2</font>, <font color="Maroon" title="c4">v3</font>, <font color="Maroon" title="c5">v4</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <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> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> &amp; <font color="Olive" title="b1">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> ) holds <br/>( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">e</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="c6">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font> &amp; <font color="Maroon" title="c6">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</font> &amp; not <!--HUMANRECFAILED-->( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> ) implies ( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c2">v1</font>,<font color="Maroon" title="c3">v2</font>
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c6">e</font> <a href="graph_1.html#R2" title="GRAPH_1:pred.2">joins</a> <font color="Maroon" title="c4">v3</font>,<font color="Maroon" title="c5">v4</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="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> ) or ( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> ) )</span><br/>

<a NAME="E3:43"/>
( ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> ) or ( the <a href="graph_1.html#U1" title="GRAPH_1:sel.1">Source</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v2</font> &amp; the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Target</a> of <font color="Maroon" title="c1">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v1</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="graph_1.html#D10" title="GRAPH_1:def.10">GRAPH_1:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43"/>
( ( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> ) or ( <font color="Maroon" title="c2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v4</font> &amp; <font color="Maroon" title="c3">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v3</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="graph_1.html#D10" title="GRAPH_1:def.10">GRAPH_1:def 10</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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