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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">M</font> is <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E2:30_1"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>and </b><br/><a NAME="E3:30_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font>
 ;<br/><a NAME="E2:30_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/><b>hence </b><a NAME="E3:30_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E1:30_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="rltopsp1.html#E2:30_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="rltopsp1.html#E3:30_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">Def1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:30_1"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c3">u</font>,<font color="Maroon" title="c4">w</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:30"/><i><font color="Green" title="E22">A4</font></i>: 
for <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">u</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> holds <br/> <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D1" target="_self" title="RLTOPSP1:def.1">RLTOPSP1:def 1</a><br/><b>let </b><font color="Maroon" title="c5">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:30"/><i><font color="Green" title="E23">A5</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>and </b><br/><a NAME="E5:30"/><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c4">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 ;<br/>

<a NAME="E6:30"/><i><font color="Green" title="E26">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">u</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E3:30"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<a NAME="E7:30"/>
 <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> <font color="Maroon" title="c4">w</font>,<font color="Maroon" title="c3">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E2:30"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="rltopsp1.html#E4:30"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="rltopsp1.html#E5:30"><i><font color="Green" title="E25">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c4">w</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="rltopsp1.html#E6:30"><i><font color="Green" title="E26">A8</font></i></a></i>;<br/>


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