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<b>let </b><font color="Maroon">X</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">N</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">N</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:35_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:35_1"/>
<font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">u</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">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="rusub_4.html#D9">RUSUB_4:def 9</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><b> such that </b><br/><a NAME="E7:35_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E8:35_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>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> <b>by </b><i/>;<br/><i><font color="Green">E49</font></i>: <font color="Maroon">z</font> <a href="rlvect_1.html#K4">+</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="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</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="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">u</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<font color="Maroon">v</font> <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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">v</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/><a NAME="E11:35_1"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</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> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E12:35_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K2" target="_self">-</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<a NAME="E2:35"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">N</font> <a href="rusub_4.html#K7">+</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">M</font></span>)</span> <a href="hidden.html#R1">=</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">w</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">w</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">N</font> &amp; <font color="Olive">w</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> <font color="Maroon">M</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="E3:35"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K2" target="_self">-</a> <font color="Maroon">M</font></span>)</span>
 ;<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">w</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:35"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E6:35"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E7:35"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">w</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> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">w'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E9:35"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w'</font>
 <b>and </b><br/><a NAME="E10:35"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">w'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E63</font></i>: <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">w</font> = 
<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="rlvect_1.html#K3">*</a> <font color="Maroon">w'</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">w'</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<font color="Maroon">v</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w'</font> = 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">w'</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">w</font> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">w</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T29">RLVECT_1:29</a></i>
<br/>.= 
<font color="Maroon">u</font> <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>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
;<br/>
<a NAME="E13:35"/><b>then </b>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:35"/>
<font color="Maroon">v</font> <a href="rusub_4.html#K5">+</a> <font color="Maroon">M</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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