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<span class="kw">consider </span><font color="Maroon" title="c3">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font><span class="kw"> such that </span><br/><a NAME="E2:47_1_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> iff ( <font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) )
 <span class="kw">and </span><br/><a NAME="E3:47_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="connsp_1.html#D7" target="_self" title="CONNSP_1:def.7">Def7</a></span>;<br/>
<a NAME="E4:47_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
 for <font color="Olive" title="b1">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/>
<a NAME="E5:47_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="connsp_1.html#T31" target="_self" title="CONNSP_1:th.31">Th33</a></span>;<br/>
<a NAME="E6:47_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<a NAME="E7:47_1_1"/>
 for <font color="Olive" title="b1">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Olive" title="b1">A</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47_1_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:47_1_1"/>
(  not  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c2">x</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c2">x</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:47_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E6:47_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="connsp_1.html#T26" target="_self" title="CONNSP_1:th.26">Th27</a>, <a class="ref" href="connsp_1.html#T38" target="_self" title="CONNSP_1:th.38">Th40</a></span>; <a class="txt" 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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