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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>  st <font color="Olive" title="b1">X</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Olive" title="b2">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> holds <br/>( <font color="Olive" title="b1">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Olive" title="b2">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Olive" title="b1">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Olive" title="b2">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> implies ( <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> ) )</span><br/>





<b>assume </b><a NAME="E1:88"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c2">X</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> &amp; <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a> )</span><br/>

<b>thus </b><a NAME="E2:88"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<a NAME="E1:88_1"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b3">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font></span><br/><b>let </b><font color="Maroon" title="c6">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> )</span><br/>



<b>assume </b><a NAME="E1:88_1_1"/><i><font color="Green" title="E40">A2</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">b1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:88_1_1"/><i><font color="Green" title="E41">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c8">b1</font> &amp; <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">b2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:88_1_1"/><i><font color="Green" title="E42">A4</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c10">b2</font> &amp; <font color="Maroon" title="c9">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c10">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E40">A2</font></i></a></i>;<br/>
<a NAME="E6:88_1_1"/><i><font color="Green" title="E43">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E1:88"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E3:88_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E5:88_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="mathmorp.html#D11" target="_self" title="MATHMORP:def.11">Def11</a></i>;<br/>
<a NAME="E7:88_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="#E1:88"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="#E3:88_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E5:88_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="mathmorp.html#D11" target="_self" title="MATHMORP:def.11">Def11</a></i>;<br/>
<a NAME="E8:88_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">b1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x3</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Olive" title="b2">b3</font></span>)</span> where <font color="Olive" title="b1">x3</font>, <font color="Olive" title="b2">b3</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : ( <font color="Olive" title="b1">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Olive" title="b2">b3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E6:88_1_1"><i><font color="Green" title="E43">A5</font></i></a></i>;<br/>
<a NAME="E9:88_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<a NAME="E10:88_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<a NAME="E11:88_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<a NAME="E12:88_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">x1</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c8">b1</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b2</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>;<br/>
<b>hence </b><a NAME="E13:88_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="#E3:88_1_1"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="#E5:88_1_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:88_1"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K4" title="MATHMORP:func.4">(+)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="mathmorp.html#D11" target="_self" title="MATHMORP:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<a NAME="E3:88"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><br/> for <font color="Olive" title="b3">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b3">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font></span><br/><b>let </b><font color="Maroon" title="c6">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> )</span><br/>



<b>assume </b><a NAME="E1:88_2"/><i><font color="Green" title="E40">A6</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">r</font> &amp; <font color="Maroon" title="c6">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">x1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:88_2"/><i><font color="Green" title="E41">A7</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> &amp; <font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c7">x1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">y1</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E5:88_2"/><i><font color="Green" title="E42">A8</font></i>: 
( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y1</font> &amp; <font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c8">y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 <b>by </b><i><a class="txt" href="#E1:88_2"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/>
<a NAME="E6:88_2"/>
<font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">b1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> or <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )</span><br/>

<b>assume </b><a NAME="E1:88_2_1"/>
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">b</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E3:88_2_1"/><i><font color="Green" title="E43">A9</font></i>: 
( <font color="Maroon" title="c9">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> )
 ;<br/>
<a NAME="E4:88_2_1"/><i><font color="Green" title="E44">A10</font></i>: 
<font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="mathmorp.html#K1" title="MATHMORP:func.1">+</a> <font color="Maroon" title="c4">x</font>
 
<b>by </b><i><a class="txt" href="#E3:88_2_1"><i><font color="Green" title="E43">A9</font></i></a></i>;<br/>
<a NAME="E5:88_2_1"/>
<font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b2</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">y</font></span>)</span> where <font color="Olive" title="b1">b2</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> : <font color="Olive" title="b1">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E3:88_2_1"><i><font color="Green" title="E43">A9</font></i></a></i>;<br/>
<a NAME="E6:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E1:88"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="#E1:88_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="txt" href="#E3:88_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="txt" href="#E5:88_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="#E4:88_2_1"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="mathmorp.html#D11" target="_self" title="MATHMORP:def.11">Def11</a></i>;<br/>
<a NAME="E7:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">b</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>;<br/>
<a NAME="E8:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T36" title="EUCLID:th.36">EUCLID:36</a></i>;<br/>
<a NAME="E9:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<a NAME="E10:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T54" title="EUCLID:th.54">EUCLID:54</a></i>;<br/>
<a NAME="E11:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T49" title="EUCLID:th.49">EUCLID:49</a></i>;<br/>
<a NAME="E12:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<a NAME="E13:88_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T52" title="EUCLID:th.52">EUCLID:52</a></i>;<br/>
<a NAME="E14:88_2_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>;<br/>
<b>hence </b><a NAME="E15:88_2_1"/>
<font color="Maroon" title="c9">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E3:88_2_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="ref" href="euclid.html#T33" title="EUCLID:th.33">EUCLID:33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:88_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c6">r</font></span>)</span> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:88"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K3" title="MATHMORP:func.3">(-)</a> <font color="Maroon" title="c3">B</font> is <a href="jordan1.html#V1" title="JORDAN1:attr.1">convex</a>
 <b>by </b><i><a class="ref" href="mathmorp.html#D11" target="_self" title="MATHMORP:def.11">Def11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
