<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</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="b2">If</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">If</font> &amp; not <font color="Olive" title="b2">If</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b2">If</font> holds <br/> ex <font color="Olive" title="b3">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b3">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b2">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b3">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</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">If</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">If</font> &amp; not <font color="Olive" title="b1">If</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">If</font> holds <br/> ex <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b2">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Olive" title="b1">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b2">B</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">If</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">If</font> &amp; not <font color="Maroon" title="c3">If</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c3">If</font> implies  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">B</font> ) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:27"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c3">If</font>
 <span class="kw">and </span><br/><a NAME="E2:27"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
not <font color="Maroon" title="c3">If</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E3:27"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c3">If</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">B</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">If</font> = <font color="Maroon" title="c3">If</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="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:27"><span class="lab"><font color="Green" title="E19">A2</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_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="E1:27_1"/>
( <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c2">A</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_1"><span class="kw">suppose </span></a><a NAME="E1:27_1_1"/>
<font color="Maroon" title="c2">A</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">B</font> )</span><br/><div class="add"><a NAME="E2:27_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">If</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T61" title="XBOOLE_1:th.61">XBOOLE_1:61</a></span>;<br/><span class="kw">hence </span><a NAME="E3:27_1_1"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">B</font> )
 ; <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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="27_1_2"><span class="kw">suppose </span></a><a NAME="E1:27_1_2"/>
not <font color="Maroon" title="c2">A</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="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Olive" title="b1">B</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">YY</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) ) </span>}</span> ;<br/><a NAME="E2:27_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) ) </span>}</span>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) ) </span>}</span> 
 ; <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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font></span><br/>

<a NAME="E2:27_1_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:27_1_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">YY</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) ) </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">V</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><span class="kw">take </span><font color="Maroon" title="c8">U</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c6">YY</font>; <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="c8">U</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font> )</span><br/><a NAME="E5:27_1_2"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">v</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">A</font> or <font color="Maroon" title="c9">v</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="c8">U</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_2"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c9">v</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">A</font>
 ; <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="c9">v</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="c8">U</font></span><br/>

<a NAME="E2:27_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</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="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/>
<a NAME="E3:27_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T8" target="_self" title="RLAFFIN2:th.8">Th8</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">IB</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:27_1_2_2"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">IB</font>
 <span class="kw">and </span><br/><a NAME="E6:27_1_2_2"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c10">IB</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="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font></span>)</span> where <font color="Olive" title="b1">B</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E8:27_1_2_2"/><span class="lab"><font color="Green" title="E25">A8</font></span>: 
<font color="Maroon" title="c10">IB</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c11">B</font>
 <span class="kw">and </span><br/><a NAME="E9:27_1_2_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_2_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a></span>;<br/>
<a NAME="E10:27_1_2_2"/>
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c11">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin2.html#E7"><span class="lab"><font color="Green" title="E5">Lm2</font></span></a></span>;<br/>
<a NAME="E11:27_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A10</font></span>: 
<font color="Maroon" title="c9">v</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="c11">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2_2"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E8:27_1_2_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a></span>;<br/>
<a NAME="E12:27_1_2_2"/>
<font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">YY</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E5:27_1_2_2"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="txt" href="#E8:27_1_2_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="txt" href="#E9:27_1_2_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a></span>;<br/>
<a NAME="E13:27_1_2_2"/><span class="kw">then </span>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:27_1_2_2"/>
<font color="Maroon" title="c9">v</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="c8">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1_2_2"><span class="lab"><font color="Green" title="E27">A10</font></span></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><span class="kw">end;</span></div><a NAME="E6:27_1_2"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<font color="Maroon" title="c8">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">U</font>
 ; <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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1_2_3"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2_3"/><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">YY</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E5:27_1_2_3"/>
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2_3"><span class="lab"><font color="Green" title="E24">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:27_1_2_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2_3"><span class="lab"><font color="Green" title="E23">A12</font></span></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><span class="kw">end;</span></div><a NAME="E7:27_1_2"/>
<font color="Maroon" title="c8">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">If</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> &amp;  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</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">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> ) );<br/>
<span class="kw">assume </span><a NAME="E1:27_1_2_4"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c8">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">If</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:27_1_2_4"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_4_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1_2_4_1"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">b</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2_4_1"/><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">YY</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_4"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E6:27_1_2_4_1"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
( <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">B</font> &amp; <font color="Maroon" title="c11">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font> &amp;  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</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">A</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c11">B</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2_4_1"><span class="lab"><font color="Green" title="E26">A17</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">B</font>
; <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="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">B</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E7:27_1_2_4_1"/>
( <font color="Maroon" title="c11">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c11">B</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2_4_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="txt" href="#E6:27_1_2_4_1"><span class="lab"><font color="Green" title="E27">A18</font></span></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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c9">p</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c4">If</font>,<span class="p1">(<span class="default"><a href="setfam_1.html#K9" title="SETFAM_1:func.9">bool</a> <font color="Maroon" title="c4">If</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:27_1_2_4"/><span class="lab"><font color="Green" title="E25">A19</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:27_1_2_4"><span class="lab"><font color="Green" title="E24">A15</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/>( $2 <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">A</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Olive" title="b1">B</font> );<br/>
<a NAME="E5:27_1_2_4"/><span class="lab"><font color="Green" title="E26">A20</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:27_1_2_4"/><span class="lab"><font color="Green" title="E27">A21</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_4_2"/><span class="lab"><font color="Green" title="E28">A22</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:27_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2_4"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">px</font> = <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:27_1_2_4_2"/><span class="lab"><font color="Green" title="E29">A23</font></span>: 
( <font color="Maroon" title="c12">y</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">A</font> &amp; <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c11">px</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2_4"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:27_1_2_4_2"><span class="lab"><font color="Green" title="E28">A22</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">y</font>
; <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="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E6:27_1_2_4_2"/>
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2_4_2"><span class="lab"><font color="Green" title="E29">A23</font></span></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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">q</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c4">If</font>,<font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E8:27_1_2_4"/><span class="lab"><font color="Green" title="E28">A24</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E6:27_1_2_4"><span class="lab"><font color="Green" title="E27">A21</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">R</font> =  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c10">q</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/>
<a NAME="E10:27_1_2_4"/><span class="lab"><font color="Green" title="E29">A25</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> or <font color="Maroon" title="c12">y</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">A</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_4_3"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ; <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="c12">y</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">A</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:27_1_2_4_3"/><span class="lab"><font color="Green" title="E30">A26</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">q</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2_4_3"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:27_1_2_4_3"/>
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2_4"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E3:27_1_2_4_3"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">px</font> = <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E7:27_1_2_4_3"/>
<font color="Maroon" title="c14">px</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E8:27_1_2_4_3"/>
<font color="Maroon" title="c12">y</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">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_2_4"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E3:27_1_2_4_3"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E4:27_1_2_4_3"><span class="lab"><font color="Green" title="E31">A27</font></span></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><span class="kw">end;</span></div>
<a NAME="E11:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A28</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E12:27_1_2_4"/><span class="lab"><font color="Green" title="E31">A29</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1_2_4"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="ref" href="convex1.html#T30" title="CONVEX1:th.30">CONVEX1:30</a></span>;<br/>
<a NAME="E13:27_1_2_4"/><span class="lab"><font color="Green" title="E32">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A31</font></span>: <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="c11">R</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="c11">R</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">R</font></span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></span>
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">L</font> being    <a href="rlvect_2.html#M2" title="RLVECT_2:mode.2">Linear_Combination</a> of <font color="Maroon" title="c11">R</font><span class="kw"> such that </span><br/><a NAME="E16:27_1_2_4"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">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="c11">R</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="c11">R</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:27_1_2_4"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
 <a href="rlaffin1.html#K3" title="RLAFFIN1:func.3">sum</a> <font color="Maroon" title="c12">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="c11">R</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="c11">R</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:27_1_2_4"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c12">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="c11">R</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="c11">R</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/>
<span class="kw">set </span><font color="Maroon" title="c13">SL</font> =  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">L</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">SLIf</font> = <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font>;<br/>
<a NAME="E19:27_1_2_4"/>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:27_1_2_4"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="rlaffin2.html#D2" target="_self" title="RLAFFIN2:def.2">Def2</a></span>;<br/>
<a NAME="E20:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A35</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">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="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T16" target="_self" title="RLAFFIN2:th.16">Th16</a></span>;<br/>
<a NAME="E21:27_1_2_4"/><span class="lab"><font color="Green" title="E38">A36</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="c11">R</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="c11">R</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> implies <font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c11">R</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:27_1_2_4_5"/><span class="lab"><font color="Green" title="E40">A38</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ; <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="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c11">R</font></span>)</span></span><br/><span class="kw">hence </span><font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">R</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="c11">R</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:27_1_2_4"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E21:27_1_2_4"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</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="c11">R</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_4_5"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
;<br/> <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><span class="kw">end;</span></div>
<a NAME="E23:27_1_2_4"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_4_6"/><span class="lab"><font color="Green" title="E41">A40</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 ; <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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font></span><br/>

<a NAME="E2:27_1_2_4_6"/><span class="kw">then </span>
<font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27_1_2_4"><span class="lab"><font color="Green" title="E39">A37</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:27_1_2_4_6"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_4_6"><span class="lab"><font color="Green" title="E41">A40</font></span></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><span class="kw">end;</span></div>
<a NAME="E24:27_1_2_4"/><span class="lab"><font color="Green" title="E41">A41</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a></span>;<br/>
<a NAME="E25:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A42</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1_2_4"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E26:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A43</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T30" title="CONVEX1:th.30">CONVEX1:30</a></span>;<br/>
<a NAME="E27:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A44</font></span>: 
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">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="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:27_1_2_4"><span class="lab"><font color="Green" title="E37">A35</font></span></a></span>;<br/>
<a NAME="E28:27_1_2_4"/><span class="lab"><font color="Green" title="E45">A45</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:27_1_2_4"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="txt" href="#E24:27_1_2_4"><span class="lab"><font color="Green" title="E41">A41</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E29:27_1_2_4"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E30:27_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A46</font></span>: 
<font color="Maroon" title="c11">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:27_1_2_4"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E31:27_1_2_4"/><span class="lab"><font color="Green" title="E47">A47</font></span>: 
<font color="Maroon" title="c4">If</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:27_1_2_4_7"/><span class="lab"><font color="Green" title="E48">A48</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">If</font>
 ; <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="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c17">G</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E3:27_1_2_4_7"/><span class="lab"><font color="Green" title="E49">A49</font></span>: 
<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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c16">F</font>
 <span class="kw">and </span><br/><a NAME="E4:27_1_2_4_7"/><span class="lab"><font color="Green" title="E50">A50</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">F</font>
 <span class="kw">and </span><br/><a NAME="E5:27_1_2_4_7"/>
<font color="Maroon" title="c17">G</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">and </span><br/><a NAME="E6:27_1_2_4_7"/><span class="lab"><font color="Green" title="E51">A51</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c17">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <font color="Maroon" title="c12">L</font>
 <span class="kw">and </span><br/><a NAME="E7:27_1_2_4_7"/><span class="lab"><font color="Green" title="E52">A52</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font> holds <br/><font color="Maroon" title="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:27_1_2_4"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E17:27_1_2_4"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E25:27_1_2_4"><span class="lab"><font color="Green" title="E42">A42</font></span></a>, <a class="ref" href="rlaffin2.html#T3" target="_self" title="RLAFFIN2:th.3">Th3</a></span>;<br/>
<a NAME="E8:27_1_2_4_7"/><span class="lab"><font color="Green" title="E53">A53</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c9">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2_4"><span class="lab"><font color="Green" title="E26">A20</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">px</font> = <font color="Maroon" title="c9">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E10:27_1_2_4_7"/><span class="lab"><font color="Green" title="E54">A54</font></span>: 
 <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c18">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="rlaffin2.html#E7"><span class="lab"><font color="Green" title="E5">Lm2</font></span></a></span>;<br/>
<a NAME="E11:27_1_2_4_7"/><span class="lab"><font color="Green" title="E55">A55</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_2_4"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a></span>;<br/>
<a NAME="E12:27_1_2_4_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A56</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</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="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1_2_4_7"><span class="lab"><font color="Green" title="E54">A54</font></span></a></span>;<br/>
<a NAME="E13:27_1_2_4_7"/><span class="lab"><font color="Green" title="E57">A57</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2_4"><span class="lab"><font color="Green" title="E25">A19</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a></span>;<br/>
<a NAME="E14:27_1_2_4_7"/><span class="lab"><font color="Green" title="E58">A58</font></span>: 
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c18">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin1.html#T65" title="RLAFFIN1:th.65">RLAFFIN1:65</a></span>;<br/>
<a NAME="E15:27_1_2_4_7"/><span class="lab"><font color="Green" title="E59">A59</font></span>: 
<font color="Maroon" title="c18">px</font> is  <a href="rlaffin1.html#V1" title="RLAFFIN1:attr.1">affinely-independent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_2_4_7"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="rlaffin1.html#T43" title="RLAFFIN1:th.43">RLAFFIN1:43</a></span>;<br/>
<a NAME="E16:27_1_2_4_7"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c18">px</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:27_1_2_4_7"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="txt" href="#E14:27_1_2_4_7"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/>
<a NAME="E17:27_1_2_4_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A60</font></span>: 
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c18">px</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">px</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:27_1_2_4_7"><span class="lab"><font color="Green" title="E54">A54</font></span></a>, <a class="txt" href="#E11:27_1_2_4_7"><span class="lab"><font color="Green" title="E55">A55</font></span></a>, <a class="txt" href="#E15:27_1_2_4_7"><span class="lab"><font color="Green" title="E59">A59</font></span></a>, <a class="ref" href="rlaffin2.html#T11" target="_self" title="RLAFFIN2:th.11">Th11</a>, <a class="ref" href="rlaffin1.html#T71" title="RLAFFIN1:th.71">RLAFFIN1:71</a></span>;<br/>
<a NAME="E18:27_1_2_4_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c18">px</font> <a href="rlvect_2.html#R1" title="RLVECT_2:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_2_4_7"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="txt" href="#E12:27_1_2_4_7"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="txt" href="#E14:27_1_2_4_7"><span class="lab"><font color="Green" title="E58">A58</font></span></a>, <a class="ref" href="rlaffin1.html#T77" title="RLAFFIN1:th.77">RLAFFIN1:77</a></span>;<br/>
<a NAME="E19:27_1_2_4_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A61</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27_1_2_4_7"><span class="lab"><font color="Green" title="E57">A57</font></span></a>, <a class="txt" href="#E17:27_1_2_4_7"><span class="lab"><font color="Green" title="E60">A60</font></span></a>, <a class="ref" href="rlvect_2.html#T28" title="RLVECT_2:th.28">RLVECT_2:28</a></span>;<br/>
<a NAME="E20:27_1_2_4_7"/>
 <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c18">px</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:27_1_2_4_7"><span class="lab"><font color="Green" title="E53">A53</font></span></a>, <a class="ref" href="rlaffin1.html#T3" title="RLAFFIN1:th.3">RLAFFIN1:3</a></span>;<br/>
<a NAME="E21:27_1_2_4_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A62</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="txt" href="#E12:27_1_2_4_7"><span class="lab"><font color="Green" title="E56">A56</font></span></a>, <a class="ref" href="rlaffin1.html#T71" title="RLAFFIN1:th.71">RLAFFIN1:71</a></span>;<br/>
<a NAME="E22:27_1_2_4_7"/><span class="lab"><font color="Green" title="E63">A63</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:27_1_2_4"><span class="lab"><font color="Green" title="E32">A30</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E23:27_1_2_4_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A64</font></span>: 
<font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c11">R</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27_1_2_4"><span class="lab"><font color="Green" title="E39">A37</font></span></a></span>;<br/>
<a NAME="E24:27_1_2_4_7"/><span class="lab"><font color="Green" title="E65">A65</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:27_1_2_4_7"><span class="lab"><font color="Green" title="E50">A50</font></span></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E66">A66</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_2_4_7_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c19">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font> implies  <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="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> )</span><br/><span class="kw">assume </span><a NAME="E1:27_1_2_4_7_1"/><span class="lab"><font color="Green" title="E67">A67</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span><br/><a NAME="E2:27_1_2_4_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:27_1_2_4"><span class="lab"><font color="Green" title="E46">A46</font></span></a>, <a class="txt" href="#E6:27_1_2_4_7"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E24:27_1_2_4_7"><span class="lab"><font color="Green" title="E65">A65</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E3:27_1_2_4_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A68</font></span>: 
( <font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:27_1_2_4"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="txt" href="#E28:27_1_2_4"><span class="lab"><font color="Green" title="E45">A45</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></a>, <a class="ref" href="rlaffin1.html#T71" title="RLAFFIN1:th.71">RLAFFIN1:71</a></span>;<br/><a NAME="E4:27_1_2_4_7_1"/>
<font color="Maroon" title="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">m</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_2_4_7"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="txt" href="#E1:27_1_2_4_7_1"><span class="lab"><font color="Green" title="E67">A67</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:27_1_2_4_7_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="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2_4_7_1"><span class="lab"><font color="Green" title="E68">A68</font></span></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><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c19">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E27:27_1_2_4_7"/><span class="lab"><font color="Green" title="E67">A69</font></span>: 
<font color="Maroon" title="c19">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">G</font>
 <span class="kw">and </span><br/><a NAME="E28:27_1_2_4_7"/><span class="lab"><font color="Green" title="E68">A70</font></span>: 
<font color="Maroon" title="c17">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:27_1_2_4"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="txt" href="#E6:27_1_2_4_7"><span class="lab"><font color="Green" title="E51">A51</font></span></a>, <a class="txt" href="#E22:27_1_2_4_7"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E29:27_1_2_4_7"/>
<font color="Maroon" title="c16">F</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c19">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">L</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:27_1_2_4_7"><span class="lab"><font color="Green" title="E52">A52</font></span></a>, <a class="txt" href="#E24:27_1_2_4_7"><span class="lab"><font color="Green" title="E65">A65</font></span></a>, <a class="txt" href="#E27:27_1_2_4_7"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="txt" href="#E28:27_1_2_4_7"><span class="lab"><font color="Green" title="E68">A70</font></span></a></span>;<br/>
<a NAME="E30:27_1_2_4_7"/><span class="kw">then </span>
<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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27_1_2_4_7"><span class="lab"><font color="Green" title="E49">A49</font></span></a>, <a class="txt" href="#E19:27_1_2_4_7"><span class="lab"><font color="Green" title="E61">A61</font></span></a>, <a class="txt" href="#E21:27_1_2_4_7"><span class="lab"><font color="Green" title="E62">A62</font></span></a>, <a class="txt" href="#E23:27_1_2_4_7"><span class="lab"><font color="Green" title="E64">A64</font></span></a>, <a class="txt" href="#E24:27_1_2_4_7"><span class="lab"><font color="Green" title="E65">A65</font></span></a>, <a class="txt" href="#E25:27_1_2_4_7"><span class="lab"><font color="Green" title="E66">A66</font></span></a>, <a class="txt" href="#E27:27_1_2_4_7"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="ref" href="rvsum_1.html#T115" title="RVSUM_1:th.115">RVSUM_1:115</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:27_1_2_4_7"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:27_1_2_4_7"><span class="lab"><font color="Green" title="E48">A48</font></span></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><span class="kw">end;</span></div>
<a NAME="E32:27_1_2_4"/>
 <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_2.html#D8" title="RLVECT_2:def.8">RLVECT_2:def 8</a></span>;<br/>
<a NAME="E33:27_1_2_4"/><span class="kw">then </span>
(  <a href="rlvect_2.html#K3" title="RLVECT_2:func.3">Carrier</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">If</font> &amp; <span class="p1">(<span class="default"><a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</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="#E31:27_1_2_4"><span class="lab"><font color="Green" title="E47">A47</font></span></a>, <a class="txt" href="#E20:27_1_2_4"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E26:27_1_2_4"><span class="lab"><font color="Green" title="E43">A43</font></span></a>, <a class="ref" href="rlaffin1.html#T71" title="RLAFFIN1:th.71">RLAFFIN1:71</a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E34:27_1_2_4"/><span class="kw">then </span>
(  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c4">If</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rlaffin1.html#K4" title="RLAFFIN1:func.4">Affin</a> <font color="Maroon" title="c4">If</font> &amp;  <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</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="Maroon" title="c12">L</font></span>)</span> <a href="rlaffin1.html#K5" title="RLAFFIN1:func.5">|--</a> <font color="Maroon" title="c4">If</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c4">If</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin2.html#T12" target="_self" title="RLAFFIN2:th.12">Th12</a>, <a class="ref" href="rlaffin1.html#T65" title="RLAFFIN1:th.65">RLAFFIN1:65</a></span>;<br/>
<a NAME="E35:27_1_2_4"/><span class="kw">then </span>
 <a href="rlvect_2.html#K6" title="RLVECT_2:func.6">Sum</a> <font color="Maroon" title="c12">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rlaffin2.html#K1" title="RLAFFIN2:func.1">Int</a> <font color="Maroon" title="c4">If</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:27_1_2_4"><span class="lab"><font color="Green" title="E44">A44</font></span></a>, <a class="ref" href="rlaffin1.html#D7" title="RLAFFIN1:def.7">RLAFFIN1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E36:27_1_2_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:27"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E12:27_1_2_4"><span class="lab"><font color="Green" title="E31">A29</font></span></a>, <a class="txt" href="#E20:27_1_2_4"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</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><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E8:27_1_2"/>
( <font color="Maroon" title="c8">U</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c3">If</font> &amp;  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="convex1.html#K3" title="CONVEX1:func.3">conv</a> <font color="Maroon" title="c8">U</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:27_1_2"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E6:27_1_2"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
