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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>) <b>-&gt; </b>   <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>  = <font color="Maroon" title="c2">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1</span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <b> such that </b><br/><a NAME="E2:34_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">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>assume </b><a NAME="E1:34_1_1_1"/>
not <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="convex4.html#K1" title="CONVEX4:func.1">Carrier</a> <font color="Maroon" title="c3">L</font>
 ;<br/><a NAME="E2:34_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 ;<br/><b>hence </b><font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">v</font> = 
<font color="Maroon" title="c2">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
<b>by </b><i><a class="txt" href="convex4.html#E2:34_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 

;<br/><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f</font> as    <a href="convex4.html#M1" title="CONVEX4:mode.1">C_Linear_Combination</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="convex4.html#D1" target="_self" title="CONVEX4:def.1">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">f</font>
;<br/>

<b>let </b><font color="Maroon" title="c7">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>thus </b><a NAME="E6:34_1_1"/>
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">v</font></span>)</span>
 <b>by </b><i><a class="txt" href="convex4.html#E2:34_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:34_1_1"/>
verum
 ;<br/>


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