<?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">M</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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Olive" title="b1">M</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</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">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Olive" title="b1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span></span><br/>





<span class="kw">thus </span><a NAME="E1:21"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1_1"/>
( <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:21_1_1_1_1"/>
( <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1_1"/>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:21_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">V</font>
 ;<br/><a NAME="E3:21_1_1_1_1_1"/><span class="kw">then </span>
 <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circled1.html#T14" target="_self" title="CIRCLED1:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:21_1_1_1_1_1"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T33" title="CONVEX1:th.33">CONVEX1:33</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:21_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="convex1.html#T34" title="CONVEX1:th.34">CONVEX1:34</a></span>;<br/><a NAME="E3:21_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circled1.html#T11" target="_self" title="CIRCLED1:th.11">Th11</a></span>;<br/><a NAME="E4:21_1_1_1_1_2"/>
 <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1_1_2"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="circled1.html#T11" target="_self" title="CIRCLED1:th.11">Th11</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>;<br/><a NAME="E5:21_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="convex1.html#T34" title="CONVEX1:th.34">CONVEX1:34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:21_1_1_1_1_2"/>
(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:21_1_1_1_1_2"><span class="lab"><font color="Green" title="E14">A3</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:21_1_1_2"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:21_1_1_2"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">v</font></span>)</span> where <font color="Olive" title="b1">v</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#D1" title="CONVEX1:def.1">CONVEX1:def 1</a></span>;<br/><span class="kw">assume </span><a NAME="E3:21_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">v</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E5:21_1_1_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font>
 <span class="kw">and </span><br/><a NAME="E6:21_1_1_2"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:21_1_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a></span>;<br/><a NAME="E7:21_1_1_2"/>
for <font color="Olive" title="b1">W</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K1" title="CIRCLED1:func.1">Circled-Family</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> holds <br/><font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">W</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="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K1" title="CIRCLED1:func.1">Circled-Family</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> implies <font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:21_1_1_2_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c6">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K1" title="CIRCLED1:func.1">Circled-Family</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</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="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">W</font> = <font color="Maroon" title="c6">W</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E3:21_1_1_2_1"/>
<font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="circled1.html#D2" target="_self" title="CIRCLED1:def.2">Def2</a></span>;<br/>
<a NAME="E4:21_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></span>;<br/>
<a NAME="E5:21_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T37" title="CONVEX1:th.37">CONVEX1:37</a></span>;<br/>
<a NAME="E6:21_1_1_2_1"/><span class="kw">then </span>
1 <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>;<br/>
<a NAME="E7:21_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="convex1.html#T32" title="CONVEX1:th.32">CONVEX1:32</a></span>;<br/>
<a NAME="E8:21_1_1_2_1"/>
<font color="Maroon" title="c7">W</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="circled1.html#D2" target="_self" title="CIRCLED1:def.2">Def2</a></span>;<br/>
<a NAME="E9:21_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circled1.html#T2" target="_self" title="CIRCLED1:th.2">Th2</a></span>;<br/>
<a NAME="E10:21_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K1" title="CIRCLED1:func.1">Circled-Family</a> <font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21_1_1_2_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="circled1.html#D2" target="_self" title="CIRCLED1:def.2">Def2</a></span>;<br/>
<a NAME="E11:21_1_1_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Olive" title="b1">w</font></span>)</span> where <font color="Olive" title="b1">w</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> </span>}</span>  &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c7">W</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21_1_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="convex1.html#D1" title="CONVEX1:def.1">CONVEX1:def 1</a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">w</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">V</font><span class="kw"> such that </span><br/><a NAME="E13:21_1_1_2_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
 <span class="kw">and </span><br/><a NAME="E14:21_1_1_2_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c8">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font>
 ;<br/>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:21_1_1_2_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<font color="Maroon" title="c8">w</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:21_1_1_2_1"/>
<font color="Maroon" title="c3">r</font> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">W</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21_1_1_2_1"><span class="lab"><font color="Green" title="E19">A11</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">hence </span><a NAME="E8:21_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:21_1_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</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><span class="kw">end;</span></div>

<a NAME="E2:21"/>
( <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> &amp; <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span> is  <a href="rltopsp1.html#V3" title="RLTOPSP1:attr.3">circled</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circled1.html#T2" target="_self" title="CIRCLED1:th.2">Th2</a>, <a class="ref" href="circled1.html#T11" target="_self" title="CIRCLED1:th.11">Th11</a>, <a class="ref" href="convex1.html#T39" title="CONVEX1:th.39">CONVEX1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:21"/>
 <a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">r</font> <a href="convex1.html#K1" title="CONVEX1:func.1">*</a> <span class="p1">(<span class="default"><a href="circled1.html#K2" title="CIRCLED1:func.2">Cir</a> <font color="Maroon" title="c2">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="circled1.html#T12" target="_self" title="CIRCLED1:th.12">Th12</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>
