<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="rltopsp1.html#NM1">LinearTopSpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>;<br/>



<a NAME="E1:108"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>;<br/>
<b>consider </b><font color="Maroon">V1</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>, <font color="Maroon">V2</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:108"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">V1</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, <a class="txt" href="rltopsp1.html#E3:108"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">U</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span>;<br/>
<a NAME="E4:108"/>
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E32:108"><i><font color="Green">E60</font></i></a></i>;<br/>
<a NAME="E5:108"/><b>then </b>
( <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span> is <a href="pre_topc.html#V3">open</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E6:108"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>
<a NAME="E7:108"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E8:108"/>
(  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> &amp;  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E9:108"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</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">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:108"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V1</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E11:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E12:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E13:108"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:108"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V2</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <font color="Maroon">V2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E16:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E17:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E13"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E18:108"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="rltopsp1.html#T10" target="_self">Th10</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">U</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> as  <a href="pre_topc.html#V3">open</a>  <a href="connsp_2.html#M1">a_neighborhood</a> of  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="rltopsp1.html#T9" target="_self">Th9</a>, <a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>take </b>
<font color="Maroon">U</font>
;<br/>

<a NAME="E20:108"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="convex1.html#T32">CONVEX1:32</a></i>;<br/>
<a NAME="E21:108"/><b>then </b><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> &amp; <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="convex1.html#T37">CONVEX1:37</a></i>;<br/>
<b>thus </b><font color="Maroon">U</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rltopsp1.html#T11" target="_self">Th11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p4">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="rltopsp1.html#K2" target="_self">-</a> <font color="Maroon">U</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D5" target="_self">RLTOPSP1:def 5</a><br/>

<b>let </b><font color="Maroon">x</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/>

<b>assume </b><a NAME="E23:108"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">U</font>
 ;<br/>

<a NAME="E24:108"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <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> <font color="Maroon">U</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">u</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>, <font color="Maroon">v</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E26:108"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E27:108"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U</font> )
 ;<br/>
<a NAME="E28:108"/>
<font color="Maroon">U</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="convex1.html#K1">*</a> <span class="p3">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E29:108"/><b>then </b>
( <font color="Maroon">u</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">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span> &amp; <font color="Maroon">v</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">V1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:108"/><b>then </b>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E31:108"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <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="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> &amp; <font color="Olive">v'</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> ) </span>}</span> 
 
;<br/>
<a NAME="E32:108"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</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">V1</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">V2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>
<a NAME="E33:108"/>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> &amp;  <a href="tops_1.html#K1">Int</a> <font color="Maroon">V2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V2</font> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E34:108"/><b>then </b>
<span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">V1</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">V2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V1</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V2</font>
 
<b>by </b><i/>;<br/>
<a NAME="E35:108"/><b>then </b>
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V1</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">V2</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E36:108"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T8" target="_self">Th8</a>, </i>;<br/>


</div>
