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<div class="add">

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</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">M</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> holds <br/> ( <font color="Olive" title="b1">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> iff for <font color="Olive" title="b2">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b3">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="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 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b2">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Olive" title="b1">M</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">M</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> iff for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font> )</span><br/>



<a NAME="E1:79"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> implies for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</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">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font></span><br/>

<b>let </b><font color="Maroon" title="c3">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="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 ) implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font> )</span><br/>

<b>assume </b><a NAME="E2:79_1"/><i><font color="Green" title="E49">A3</font></i>: 
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="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 )
 ; <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="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font></span><br/>

<a NAME="E3:79_1"/>
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )</span><br/>

<b>assume </b><a NAME="E1:79_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>then consider </b><font color="Maroon" title="c5">u</font>, <font color="Maroon" title="c6">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:79_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">u</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c6">v</font> &amp; <font color="Maroon" title="c5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> )
 ;<br/>
<a NAME="E4:79_1_1"/><i><font color="Green" title="E51">A5</font></i>: 
 ex <font color="Olive" title="b1">w1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c5">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">w1</font> &amp; <font color="Olive" title="b1">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="txt" href="convex4.html#E3:79_1_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<a NAME="E5:79_1_1"/>
 ex <font color="Olive" title="b1">w2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">w2</font> &amp; <font color="Olive" title="b1">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="txt" href="convex4.html#E3:79_1_1"><i><font color="Green" title="E50">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:79_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex4.html#E1:79_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="convex4.html#E2:79_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="convex4.html#E3:79_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="convex4.html#E4:79_1_1"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="convex4.html#D23" target="_self" title="CONVEX4:def.23">Def202</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="E4:79_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c3">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</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="E2:79"/>
( ( for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font> ) implies <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_2"/><i><font color="Green" title="E48">A7</font></i>: 
for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a></span><br/>

<b>let </b><font color="Maroon" title="c3">u</font>, <font color="Maroon" title="c4">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="convex4.html#D23" target="_self" title="CONVEX4:def.23">CONVEX4:def 23</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">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) &amp; <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font></span><br/><b>let </b><font color="Maroon" title="c5">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="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="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )</span><br/>



<b>assume </b><a NAME="E2:79_2"/>
 ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="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 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> or  not <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> )</span><br/>

<a NAME="E3:79_2"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font>
 
<b>by </b><i><a class="txt" href="convex4.html#E1:79_2"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:79_2"/>
( <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</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="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E5:79_2"/><b>then </b>
( <font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> &amp; <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> )
 
;<br/>
<a NAME="E6:79_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:79_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c3">u</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex4.html#E3:79_2"><i><font color="Green" title="E49">A11</font></i></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="E3:79"/>
( <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> iff for <font color="Olive" title="b1">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b2">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">r</font> &amp; <font color="Olive" title="b2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K6" title="RUSUB_4:func.6">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6" title="COMPLEX1:func.6">1r</a>  <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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="c2">M</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:79"><i><font color="Green" title="E47">A1</font></i></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>
