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<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#K5" title="MATHMORP:func.5">(O)</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#K6" title="MATHMORP:func.6">(o)</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#K5" title="MATHMORP:func.5">(O)</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#K6" title="MATHMORP:func.6">(o)</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="E1:89"/><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#K5" title="MATHMORP:func.5">(O)</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#K6" title="MATHMORP:func.6">(o)</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="E2:89"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</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#K6" title="MATHMORP:func.6">(o)</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="89_1"><b>proof </b></a><div class="add">

<a NAME="E1:89_1"/>
<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="txt" href="mathmorp.html#E1:89"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="mathmorp.html#T77" target="_self" title="MATHMORP:th.77">Th77</a></i>;<br/>
<b>hence </b><a NAME="E2:89_1"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K5" title="MATHMORP:func.5">(O)</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="txt" href="mathmorp.html#E1:89"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="mathmorp.html#T77" target="_self" title="MATHMORP:th.77">Th77</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="E3:89"/>
<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="txt" href="mathmorp.html#E1:89"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="mathmorp.html#T77" target="_self" title="MATHMORP:th.77">Th77</a></i>;<br/>
<b>hence </b><a NAME="E4:89"/>
<font color="Maroon" title="c2">X</font> <a href="mathmorp.html#K6" title="MATHMORP:func.6">(o)</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="txt" href="mathmorp.html#E1:89"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="mathmorp.html#T77" target="_self" title="MATHMORP:th.77">Th77</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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