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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">M1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>, <font color="Maroon">M2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">r1</font> be   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">r2</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:6"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">M1</font> is <a href="rltopsp1.html#V3">circled</a>
 <b>and </b><br/><a NAME="E2:6"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">M2</font> is <a href="rltopsp1.html#V3">circled</a>
 ;<br/>

<b>let </b><font color="Maroon">u</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="circled1.html#D1" target="_self">CIRCLED1:def 1</a><br/><b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E3:6"/><i><font color="Green">E38</font></i>: 
 <a href="complex1.html#K18">abs</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E4:6"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>

<a NAME="E5:6"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</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">u1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">u2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E7:6"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font> &amp; <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E8:6"/>
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E10:6"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> )
 ;<br/>
<a NAME="E11:6"/>
<font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <font color="Olive">x</font></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="convex1.html#D1">CONVEX1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E13:6"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E14:6"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> )
 
<b>by </b><i>, , , , , </i>;<br/>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E16:6"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T1">RLTOPSP1:1</a></i>;<br/>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E18:6"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</font>
 
<b>by </b><i>, <a class="ref" href="rltopsp1.html#T1">RLTOPSP1:1</a></i>;<br/>
<a NAME="E19:6"/>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>;<br/>
<a NAME="E20:6"/><b>then </b>
<font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">y</font></span>)</span> where <font color="Olive">x</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Olive">y</font> is   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E21:6"/>
<font color="Maroon">p</font> <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"><font color="Maroon">r1</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M1</font></span>)</span> <a href="rusub_4.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="convex1.html#K1">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rusub_4.html#D10">RUSUB_4:def 10</a></i>;<br/>


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