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

<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Af</font> being   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st  not <font color="Olive" title="b1">Af</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  holds <br/><span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">Af</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">Af</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Af</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">Af</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies <span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">Af</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">Af</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:20"/>
 not <font color="Maroon" title="c2">Af</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">Af</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">Af</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">a</font> = <font color="Maroon" title="c2">Af</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">L</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c2">Af</font><span class="kw"> such that </span><br/><a NAME="E4:20"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K2" title="RLVECT_2:func.2">Sum</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:20"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:20"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Af</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T15" target="_self" title="RLAFFIN2:th.15">Th15</a></span>;<br/>
<a NAME="E7:20"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Af</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Af</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E8:20"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
 for <font color="Olive" title="b1">v</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> holds   <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:20_1_1"/>
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Af</font> or  not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Af</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_1"/><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Af</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font></span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">Af</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="funct_4.html#T13" title="FUNCT_4:th.13">FUNCT_4:13</a></span>
<br/>.= 
1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c3">a</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:20_1_1_1"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_1"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:20_1_1_2"/>
 not <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Af</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font></span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><a href="rlvect_2.html#K4" title="RLVECT_2:func.4">ZeroLC</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:20"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E7:20"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="funct_4.html#T11" title="FUNCT_4:th.11">FUNCT_4:11</a></span>
<br/>.= 
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#T20" title="RLVECT_2:th.20">RLVECT_2:20</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:20_1_1_2"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E9:20"/>
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:20"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="xcmplx_1.html#T87" title="XCMPLX_1:th.87">XCMPLX_1:87</a></span>;<br/>
<a NAME="E10:20"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c4">L</font> is  <a href="convex1.html#V2" title="CONVEX1:attr.2">convex</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:20"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="rlaffin1.html#T62" title="RLAFFIN1:th.62">RLAFFIN1:62</a></span>;<br/>
<a NAME="E11:20"/><span class="kw">then </span>
<font color="Maroon" title="c4">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#D1" title="CONVEX3:def.1">CONVEX3:def 1</a></span>;<br/>
<a NAME="E12:20"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Olive" title="b1">K</font></span>)</span> where <font color="Olive" title="b1">K</font> is   <a href="convex2.html#NM2" title="CONVEX2:NM.2">Convex_Combination</a> of <font color="Maroon" title="c3">a</font> : <font color="Olive" title="b1">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex3.html#K1" title="CONVEX3:func.1">ConvexComb</a> <font color="Maroon" title="c1">V</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:20"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>;<br/>
<a NAME="E13:20"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">Af</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex3.html#T5" title="CONVEX3:th.5">CONVEX3:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:20"/>
<span class="p1">(<span class="default"><a href="rlaffin2.html#K2" title="RLAFFIN2:func.2">center_of_mass</a> <font color="Maroon" title="c1">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">Af</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">Af</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:20"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="rlaffin2.html#D2" target="_self" title="RLAFFIN2:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
