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<span class="kw">set </span><font color="Maroon" title="c5">IT</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">ss</font> where <font color="Olive" title="b1">ss</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> </span>}</span> ;<br/>
<a NAME="E1:52_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">ss</font> where <font color="Olive" title="b1">ss</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ss</font> where <font color="Olive" title="b1">ss</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> </span>}</span>  or <font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">ss</font> where <font color="Olive" title="b1">ss</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> </span>}</span> 
 ; <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="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span><br/>

<a NAME="E2:52_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">ss</font> being   <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  st <br/>( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:52_1_1_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:52_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">ss</font> where <font color="Olive" title="b1">ss</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  : <font color="Olive" title="b1">ss</font> <a href="sgraph1.html#R3" title="SGRAPH1:pred.3">is_path_of</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c4">v2</font> </span>}</span>  is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</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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