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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="rltopsp1.html#V3">circled</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">B</font> be  <a href="rltopsp1.html#V3">circled</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</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">RLTOPSP1:def 6</a><br/>





<b>assume </b><a NAME="E1:2"/><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="E2:2"/><i><font color="Green">E37</font></i>: 
<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_5.html#K1">-</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">v</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_5.html#K1">-</a> <font color="Maroon">B</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/>
<a NAME="E3:2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="rusub_5.html#K1">-</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#K6">-</a> <font color="Olive">v</font></span>)</span> where <font color="Olive">u</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Olive">v</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</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_5.html#D2">RUSUB_5:def 2</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="E4:2"/>
<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_5.html#K1">-</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x'</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E6:2"/><i><font color="Green">E40</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="E7:2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_5.html#K1">-</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">u</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E9:2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">v</font>
 <b>and </b><br/><a NAME="E10:2"/><i><font color="Green">E43</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="E11:2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:2"/><i><font color="Green">E45</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>, <a class="ref" href="rltopsp1.html#D6">RLTOPSP1:def 6</a></i>;<br/>
<a NAME="E13:2"/><i><font color="Green">E46</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="ref" href="rltopsp1.html#T1">RLTOPSP1:1</a></i>;<br/>
<a NAME="E14:2"/>
<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#K6">-</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#T48">RLVECT_1:48</a></i>;<br/>
<b>hence </b><a NAME="E15:2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="rusub_5.html#K1">-</a> <font color="Maroon">B</font>
 <b>by </b><i>, , </i>;<br/>


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