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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="rltopsp1.html#V3" target="_self">circled</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">B</font> be  <a href="rltopsp1.html#V3" target="_self">circled</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rltopsp1.html#D6" target="_self">RLTOPSP1:def 6</a><br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green">E36</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<a NAME="E2:45"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">A</font> <a href="rusub_4.html#K7">+</a> <font color="Maroon">B</font> <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">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">A</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</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>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="E3:45"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</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>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:45"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E6:45"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x'</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>
 ;<br/>
<b>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="E8:45"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E9:45"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E10:45"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rltopsp1.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E11:45"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:45"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">A</font> &amp; <font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r</font> <a href="convex1.html#K1">*</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="txt" href="rltopsp1.html#E1:45"><i><font color="Green">E36</font></i></a>, <a class="txt" href="rltopsp1.html#E2:45"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E13:45"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<b>hence </b><a NAME="E14:45"/>
<font color="Maroon">x</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="txt" href="rltopsp1.html#E5:45"><i><font color="Green">E38</font></i></a>, <a class="ref" href="rltopsp1.html#T3" target="_self">Th3</a>, </i>;<br/>


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