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<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V3">Abelian</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V7">RealLinearSpace-like</a>  <a href="rlvect_1.html#L2">RLSStruct</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="E1:18"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">M1</font> is <a href="convex1.html#V1" target="_self">convex</a>
 <b>and </b><br/><a NAME="E2:18"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">M2</font> is <a href="convex1.html#V1" target="_self">convex</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>, <font color="Maroon">v</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="convex1.html#D2" target="_self">CONVEX1:def 2</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="E3:18"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E37</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" target="_self">*</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" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="convex1.html#K1" target="_self">*</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" target="_self">*</a> <font color="Maroon">M2</font></span>)</span>
 ;<br/>

<a NAME="E6:18"/>
<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" target="_self">*</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" target="_self">*</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="E8:18"/><i><font color="Green">E46</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" target="_self">*</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" target="_self">*</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E9:18"/>
<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/>;<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="E11:18"/><i><font color="Green">E47</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="E12:18"/>
<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/>;<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="E14:18"/><i><font color="Green">E51</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="E15:18"/>
<font color="Maroon">v</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" target="_self">*</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" target="_self">*</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">v1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>, <font color="Maroon">v2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E17:18"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">v1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M1</font> &amp; <font color="Maroon">v2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E18:18"/>
<font color="Maroon">v1</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/>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E20:18"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> )
 ;<br/>
<a NAME="E21:18"/>
<font color="Maroon">v2</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/>;<br/>
<b>then consider </b><font color="Maroon">y2</font> being   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E23:18"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">v2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> )
 ;<br/>
<a NAME="E24:18"/><i><font color="Green">E60</font></i>: 
( <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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M1</font> &amp; <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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">M2</font> )
 
<b>by </b><i>, , , , , , , </i>;<br/>
<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"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span>
<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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E26:18"/><b>then </b>
<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"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <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/>;<br/>
<a NAME="E27:18"/><b>then </b><i><font color="Green">E61</font></i>: 
<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"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">r1</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M1</font>
 
;<br/>
<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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">x2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/>
<a NAME="E29:18"/><b>then </b>
<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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> <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/>;<br/>
<a NAME="E30:18"/><b>then </b><i><font color="Green">E62</font></i>: 
<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> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">r2</font> <a href="convex1.html#K1" target="_self">*</a> <font color="Maroon">M2</font>
 
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u2</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6">RLVECT_1:def 6</a></i>
;<br/>
<a NAME="E32:18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">v2</font></span>)</span></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" target="_self">*</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" target="_self">*</a> <font color="Maroon">M2</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E33:18"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">u</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="convex1.html#K1" target="_self">*</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" target="_self">*</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/>


</div>
