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<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>





<a NAME="E1:115"/><i><font color="Green">E36</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Olive">v</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> : ( <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>assume </b><a NAME="E2:115"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">b</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:115"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E6:115"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E7:115"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="115_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:115_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:115_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 ;<br/><a NAME="E3:115_1"/>
<font color="Maroon">W</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="rltopsp1.html#T12" target="_self">Th12</a>, <a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/><b>then consider </b><font color="Maroon">W1</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">a</font>, <font color="Maroon">W2</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">b</font><b> such that </b><br/><a NAME="E5:115_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">W1</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">W2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E2:115"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E6:115_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E7:115_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W1</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:115_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E10:115_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E12:115_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E13:115_1"/><b>then </b>
<font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W2</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E15:115_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E16:115_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/><a NAME="E18:115_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T15" target="_self">Th15</a>, , </i>;<br/><a NAME="E19:115_1"/><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">W1</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">W2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T13" target="_self">Th13</a>, <a class="txt" href="rltopsp1.html#E6:115"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E20:115_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">W1</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">W2</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E21:115_1"/><b>then </b>
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="rltopsp1.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E22:115_1"/>
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:115"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/>


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