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<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">P</font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<a NAME="E3:12"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font> is <a href="connsp_1.html#V1">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>
<a NAME="E4:12"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 
;<br/>
<a NAME="E5:12"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
<font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T28">PRE_TOPC:28</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">Q</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/>


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