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<div class="add">

<b>set </b><font color="Maroon">FF</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> ;<br/>
<a NAME="E1:101_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:101_1_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span> 
 ;<br/><a NAME="E2:101_1_1_1_1"/><b>then </b>
 ex <font color="Olive">P</font>, <font color="Olive">S</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Olive">S</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Olive">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E3:101_1_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">H</font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><font color="Olive">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font></span>)</span></span>)</span> where <font color="Olive">H</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Olive">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> : ( <font color="Olive">H</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> &amp; <font color="Olive">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font> ) </span>}</span>  is   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">X</font>
 ;<br/>


</div>
