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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="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="clvect_1.html#V2" title="CLVECT_1:attr.2">ComplexLinearSpace-like</a>  <a href="clvect_1.html#L1" title="CLVECT_1:struct.1">CLSStruct</a> ;<br/><b>let </b><font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c3">N</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> &amp; <font color="Maroon" title="c3">N</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> )
 ;<br/>

<a NAME="E2:77"/>
for <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b3">z</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b4">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">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="b4">r</font> &amp; <font color="Olive" title="b4">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 ) &amp; <font color="Olive" title="b1">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="b3">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">u</font>, <font color="Maroon" title="c5">v</font> be   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c6">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/>



<b>assume </b><a NAME="E1:77_1"/><i><font color="Green" title="E47">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="c6">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="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font> &amp; <font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">y1</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="E3:77_1"/><i><font color="Green" title="E48">A4</font></i>: 
( <font color="Maroon" title="c4">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">y1</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">M</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">y2</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="E5:77_1"/><i><font color="Green" title="E49">A5</font></i>: 
( <font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y2</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">M</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E1:77_1"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>
<a NAME="E6:77_1"/><i><font color="Green" title="E50">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</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">M</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">N</font> )
 
<b>by </b><i><a class="txt" href="convex4.html#E1:77"><i><font color="Green" title="E46">A1</font></i></a>, <a class="txt" href="convex4.html#E1:77_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="convex4.html#E3:77_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="convex4.html#E5:77_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="convex4.html#D23" target="_self" title="CONVEX4:def.23">Def202</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="convex4.html#E3:77_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="txt" href="convex4.html#E5:77_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="clvect_1.html#D2" title="CLVECT_1:def.2">CLVECT_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y2</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">x1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c9">x2</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c10">y2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E8:77_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c4">u</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</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="c6">z</font></span>)</span> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c5">v</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font>
 <b>by </b><i><a class="txt" href="convex4.html#E6:77_1"><i><font color="Green" title="E50">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:77"/>
<font color="Maroon" title="c2">M</font> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</a> <font color="Maroon" title="c3">N</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a>
 <b>by </b><i><a class="ref" href="convex4.html#D23" target="_self" title="CONVEX4:def.23">Def202</a></i>;<br/>


</div>
