<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a>   <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a>   <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">vector-distributive</a>   <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">scalar-distributive</a>   <a href="rlvect_1.html#V7" title="RLVECT_1:attr.7">scalar-associative</a>   <a href="rlvect_1.html#V8" title="RLVECT_1:attr.8">scalar-unital</a>   <a href="rlvect_1.html#L1" title="RLVECT_1:struct.1">RLSStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M1</font>, <font color="Olive" title="b2">M2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">M1</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  &amp; <font color="Olive" title="b2">M2</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b2">M2</font></span>)</span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">M1</font>, <font color="Maroon" title="c3">M2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Maroon" title="c2">M1</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  &amp; <font color="Maroon" title="c3">M2</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">r1</font>, <font color="Maroon" title="c5">r2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">M1</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  &amp; <font color="Maroon" title="c3">M2</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a>  )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:18"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c2">M1</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 <span class="kw">and </span><br/><a NAME="E2:18"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c3">M2</font> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> is  <a href="convex1.html#V1" title="CONVEX1:attr.1">convex</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="convex1.html#D2" target="_self" title="CONVEX1:def.2">CONVEX1:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> holds <br/><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="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.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="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> &amp; <font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.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="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:18"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c8">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 )
 <span class="kw">and </span><br/><a NAME="E4:18"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:18"/><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.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="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span></span><br/>

<a NAME="E6:18"/>
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">v1</font>, <font color="Maroon" title="c10">v2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E8:18"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v2</font>
 <span class="kw">and </span><br/><a NAME="E9:18"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font>
 <span class="kw">and </span><br/><a NAME="E10:18"/><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c10">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font>
 ;<br/>
<a NAME="E11:18"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">u1</font>, <font color="Maroon" title="c12">u2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E13:18"/><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u2</font>
 <span class="kw">and </span><br/><a NAME="E14:18"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c11">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font>
 <span class="kw">and </span><br/><a NAME="E15:18"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<font color="Maroon" title="c12">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">y1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E17:18"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c9">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font>
 <span class="kw">and </span><br/><a NAME="E18:18"/><span class="lab"><font color="Green" title="E23">A13</font></span>: 
<font color="Maroon" title="c13">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:18"><span class="lab"><font color="Green" title="E17">A7</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">x1</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E20:18"/><span class="lab"><font color="Green" title="E24">A14</font></span>: 
<font color="Maroon" title="c11">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">and </span><br/><a NAME="E21:18"/><span class="lab"><font color="Green" title="E25">A15</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:18"><span class="lab"><font color="Green" title="E20">A10</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E26">A16</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:18"><span class="lab"><font color="Green" title="E24">A14</font></span></a>, <a class="txt" href="#E17:18"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</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"><font color="Maroon" title="c4">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font></span>)</span></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"><font color="Maroon" title="c4">r1</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">y2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E24:18"/><span class="lab"><font color="Green" title="E27">A17</font></span>: 
<font color="Maroon" title="c10">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font>
 <span class="kw">and </span><br/><a NAME="E25:18"/><span class="lab"><font color="Green" title="E28">A18</font></span>: 
<font color="Maroon" title="c15">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:18"><span class="lab"><font color="Green" title="E18">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">x2</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E27:18"/><span class="lab"><font color="Green" title="E29">A19</font></span>: 
<font color="Maroon" title="c12">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</font>
 <span class="kw">and </span><br/><a NAME="E28:18"/><span class="lab"><font color="Green" title="E30">A20</font></span>: 
<font color="Maroon" title="c16">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:18"><span class="lab"><font color="Green" title="E21">A11</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E31">A21</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:18"><span class="lab"><font color="Green" title="E29">A19</font></span></a>, <a class="txt" href="#E24:18"><span class="lab"><font color="Green" title="E27">A17</font></span></a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</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"><font color="Maroon" title="c5">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</font></span>)</span></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"><font color="Maroon" title="c5">r2</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
<font color="Maroon" title="c5">r2</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
;<br/>
<a NAME="E30:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c16">x2</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c15">y2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">M2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E28:18"><span class="lab"><font color="Green" title="E30">A20</font></span></a>, <a class="txt" href="#E25:18"><span class="lab"><font color="Green" title="E28">A18</font></span></a>, <a class="ref" href="convex1.html#D2" target="_self" title="CONVEX1:def.2">Def2</a></span>;<br/>
<a NAME="E31:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A22</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:18"><span class="lab"><font color="Green" title="E31">A21</font></span></a></span>;<br/>
<a NAME="E32:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c14">x1</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E13">A3</font></span></a>, <a class="txt" href="#E21:18"><span class="lab"><font color="Green" title="E25">A15</font></span></a>, <a class="txt" href="#E18:18"><span class="lab"><font color="Green" title="E23">A13</font></span></a>, <a class="ref" href="convex1.html#D2" target="_self" title="CONVEX1:def.2">Def2</a></span>;<br/>
<a NAME="E33:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A23</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span> where <font color="Olive" title="b1">x</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M1</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:18"><span class="lab"><font color="Green" title="E26">A16</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u2</font></span>)</span></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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</font></span>)</span></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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</font></span>)</span></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"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D8" title="RLVECT_1:def.8">RLVECT_1:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span></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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</font></span>)</span></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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c11">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c9">v1</font></span>)</span></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"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c12">u2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c10">v2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></span>
;<br/>
<a NAME="E35:18"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c12">u2</font></span>)</span></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="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">v2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:18"><span class="lab"><font color="Green" title="E33">A23</font></span></a>, <a class="txt" href="#E31:18"><span class="lab"><font color="Green" title="E32">A22</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:18"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c6">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.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="Maroon" title="c8">p</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r1</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M1</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r2</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c3">M2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:18"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="txt" href="#E8:18"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="rusub_4.html#D10" title="RUSUB_4:def.10">RUSUB_4:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
