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<b>let </b><font color="Maroon" title="c2">g1</font>, <font color="Maroon" title="c3">g2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ,<span class="p2">(<span class="default"><a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <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">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e</font></span>)</span> ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e</font></span>)</span> ) implies <font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )</span><br/>

<b>assume </b><a NAME="E1:56_1_2"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">g1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e</font></span>)</span>
 ; <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">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> ex <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> st not <font color="Maroon" title="c3">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e</font></span>)</span> or <font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> )</span><br/>

<b>assume </b><a NAME="E2:56_1_2"/><i><font color="Green" title="E37">A4</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/> for <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c3">g2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Olive" title="b2">e</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">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">g1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">e</font></span><br/><b>let </b><font color="Maroon" title="c5">e</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="convex4.html#K10" title="CONVEX4:func.10">C_LinComb</a> <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">g1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">e</font></span><br/><b>thus </b><font color="Maroon" title="c2">g1</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">e</font> = 
<font color="Maroon" title="c4">x</font> <a href="convex4.html#K7" title="CONVEX4:func.7">*</a> <span class="p1">(<span class="default"><a href="convex4.html#K11" title="CONVEX4:func.11">@</a> <font color="Maroon" title="c5">e</font></span>)</span>
<b>by </b><i><a class="txt" href="convex4.html#E1:56_1_2"><i><font color="Green" title="E36">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">g2</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">e</font>
<b>by </b><i><a class="txt" href="convex4.html#E2:56_1_2"><i><font color="Green" title="E37">A4</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="E4:56_1_2"/>
<font color="Maroon" title="c2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</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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