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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="clvect_1.html#NM2" title="CLVECT_1:NM.2">ComplexLinearSpace</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><br/> for <font color="Olive" title="b2">z1</font>, <font color="Olive" title="b3">z2</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b4">r1</font>, <font color="Olive" title="b5">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b2">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r1</font> &amp; <font color="Olive" title="b3">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">r2</font> &amp; <font color="Olive" title="b4">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b5">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; <font color="Olive" title="b1">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">z2</font> <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> <span class="p1">(<span class="default"><font color="Olive" title="b2">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Olive" title="b3">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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"> for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>  st  ex <font color="Olive" title="b3">r1</font>, <font color="Olive" title="b4">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">r1</font> &amp; <font color="Olive" title="b2">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">r2</font> &amp; <font color="Olive" title="b3">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b4">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z2</font> <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> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><b>let </b><font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">z2</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">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) &amp; <font color="Maroon" title="c2">M</font> is <a href="convex4.html#V2" title="CONVEX4:attr.2">convex</a> implies <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> )</span><br/>





<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E59">A1</font></i>: 
 ex <font color="Olive" title="b1">r1</font>, <font color="Olive" title="b2">r2</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r1</font> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r2</font> &amp; <font color="Olive" title="b1">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Olive" title="b2">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E60">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/>

<b>consider </b><font color="Maroon" title="c5">r1</font>, <font color="Maroon" title="c6">r2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:97"/><i><font color="Green" title="E61">B1</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">r2</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="txt" href="convex4.html#E1:97"><i><font color="Green" title="E59">A1</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_1"/>
( <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_1"><b>suppose </b></a><a NAME="E1:97_1_1"/>
<font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:97_1_1"/><b>then </b>
( <font color="Maroon" title="c3">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="ref" href="convex4.html#T63" target="_self" title="CONVEX4:th.63">Lm207</a></i>;<br/><b>hence </b><a NAME="E3:97_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="convex4.html#T64" target="_self" title="CONVEX4:th.64">Lm208</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2"><b>suppose </b></a><a NAME="E1:97_1_2"/><i><font color="Green" title="E62">A3</font></i>: 
not <font color="Maroon" title="c2">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <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">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:97_1_2_1"/>
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or ( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1_1"><b>suppose </b></a><a NAME="E1:97_1_2_1_1"/><i><font color="Green" title="E63">A4</font></i>: 
<font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <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">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:97_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c3">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="convex4.html#E1:97_1_2"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="convex4.html#T52" target="_self" title="CONVEX4:th.52">Lm202</a></i>;<br/><b>hence </b><a NAME="E3:97_1_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex4.html#E1:97_1_2_1_1"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="convex4.html#T65" target="_self" title="CONVEX4:th.65">Lm209</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1_2"><b>suppose </b></a><a NAME="E1:97_1_2_1_2"/><i><font color="Green" title="E63">A5</font></i>: 
<font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <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">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:97_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c4">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="convex4.html#E1:97_1_2"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="convex4.html#T52" target="_self" title="CONVEX4:th.52">Lm202</a></i>;<br/><b>hence </b><a NAME="E3:97_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex4.html#E1:97_1_2_1_2"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="convex4.html#T65" target="_self" title="CONVEX4:th.65">Lm209</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="97_1_2_1_3"><b>suppose </b></a><a NAME="E1:97_1_2_1_3"/><i><font color="Green" title="E63">A6</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 ; <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">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:97_1_2_1_3"/><b>then </b><i><font color="Green" title="E64">A7</font></i>: 
( <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp; <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">r1</font> &amp; <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">r2</font> )
 <b>by </b><i><a class="txt" href="convex4.html#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>;<br/><a NAME="E3:97_1_2_1_3"/><b>then </b>
(  <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="Maroon" title="c5">r1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> &amp; <font color="Maroon" title="c5">r1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &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="Maroon" title="c6">r2</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> &amp; <font color="Maroon" title="c6">r2</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="convex4.html#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="txt" href="convex4.html#E1:97_1_2_1_3"><i><font color="Green" title="E63">A6</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a>, <a class="ref" href="real_2.html#T142" title="REAL_2:th.142">REAL_2:142</a></i>;<br/><a NAME="E4:97_1_2_1_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span></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#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="txt" href="convex4.html#E2:97"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="convex4.html#T48" target="_self" title="CONVEX4:th.48">Th204</a></i>;<br/><a NAME="E5:97_1_2_1_3"/><b>then </b><i><font color="Green" title="E65">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="rusub_4.html#K7" title="RUSUB_4:func.7">+</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> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="ref" href="convex4.html#T62" target="_self" title="CONVEX4:th.62">Lm206</a></i>;<br/><i><font color="Green" title="E66">A11</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="ref" href="convex4.html#T54" target="_self" title="CONVEX4:th.54">Lm204</a></i>
<br/>.= 
<font color="Maroon" title="c3">z1</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="txt" href="convex4.html#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="txt" href="convex4.html#E2:97_1_2_1_3"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/><a NAME="E7:97_1_2_1_3"/>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E2:97_1_2_1_3"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>;<br/><a NAME="E8:97_1_2_1_3"/><b>then </b>
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">r1</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c6">r2</font></span>)</span>
 ;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</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> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">z2</font> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="txt" href="convex4.html#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="ref" href="convex4.html#T54" target="_self" title="CONVEX4:th.54">Lm204</a></i>
<br/>.= 
<font color="Maroon" title="c4">z2</font> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="txt" href="convex4.html#E4:97"><i><font color="Green" title="E61">B1</font></i></a>, <a class="txt" href="convex4.html#E2:97_1_2_1_3"><i><font color="Green" title="E64">A7</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>
;<br/><b>hence </b><a NAME="E10:97_1_2_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</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#K7" title="RUSUB_4:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">z2</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z1</font> <a href="complex1.html#K8" title="COMPLEX1:func.8">+</a> <font color="Maroon" title="c4">z2</font></span>)</span> <a href="convex4.html#K19" title="CONVEX4:func.19">*</a> <font color="Maroon" title="c2">M</font>
 <b>by </b><i><a class="txt" href="convex4.html#E5:97_1_2_1_3"><i><font color="Green" title="E65">A10</font></i></a>, <a class="txt" href="convex4.html#E6:97_1_2_1_3"><i><font color="Green" title="E66">A11</font></i></a>, <a class="ref" href="convex4.html#T55" target="_self" title="CONVEX4:th.55">Lm205</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></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

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