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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>;<br/><b>let </b><font color="Maroon" title="c2">V1</font>, <font color="Maroon" title="c3">V2</font>, <font color="Maroon" title="c4">V3</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><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c2">V1</font> is <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">linearly-closed</a> &amp; <font color="Maroon" title="c3">V2</font> is <a href="clvect_1.html#V3" title="CLVECT_1:attr.3">linearly-closed</a> )
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c4">V3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font></span>)</span> where <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> is   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> : ( <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> ) </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E3:39"/>
for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being  <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font> holds <br/><font color="Olive" title="b1">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_1.html#D4" target="_self" title="CLVECT_1:def.4">CLVECT_1:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">v</font>, <font color="Maroon" title="c6">u</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><b>that </b><br/><a NAME="E1:39_1"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
 <b>and </b><br/><a NAME="E2:39_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green" title="E22">A5</font></i>: 
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font>
 <b>and </b><br/><a NAME="E5:39_1"/><i><font color="Green" title="E23">A6</font></i>: 
( <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="clvect_1.html#E1:39_1"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">u1</font>, <font color="Maroon" title="c10">u2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E7:39_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">u1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u2</font>
 <b>and </b><br/><a NAME="E8:39_1"/><i><font color="Green" title="E25">A8</font></i>: 
( <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c10">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="clvect_1.html#E2:39_1"><i><font color="Green" title="E21">A4</font></i></a></i>;<br/>
<a NAME="E9:39_1"/><i><font color="Green" title="E26">A9</font></i>: 
( <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> )
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="clvect_1.html#E5:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="clvect_1.html#E8:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="clvect_1.html#D4" target="_self" title="CLVECT_1:def.4">Def4</a></i>;<br/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u2</font>
<b>by </b><i><a class="txt" href="clvect_1.html#E4:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="clvect_1.html#E7:39_1"><i><font color="Green" title="E24">A7</font></i></a>, <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="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u2</font>
<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"><font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c9">u1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v2</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c10">u2</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/>
<b>hence </b><a NAME="E11:39_1"/>
<font color="Maroon" title="c5">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="clvect_1.html#E9:39_1"><i><font color="Green" title="E26">A9</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c5">z</font> be   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a>;<br/><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="E4:39"/>
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">v1</font>, <font color="Maroon" title="c8">v2</font> being   <a href="clvect_1.html#NM1" title="CLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E6:39"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">v1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">v2</font>
 <b>and </b><br/><a NAME="E7:39"/><i><font color="Green" title="E21">A11</font></i>: 
( <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> )
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a></i>;<br/>
<a NAME="E8:39"/><i><font color="Green" title="E22">A12</font></i>: 
( <font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c7">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V1</font> &amp; <font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c8">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V2</font> )
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="clvect_1.html#E7:39"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="clvect_1.html#D4" target="_self" title="CLVECT_1:def.4">Def4</a></i>;<br/>
<a NAME="E9:39"/>
<font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c7">v1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <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="c8">v2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E6:39"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="clvect_1.html#D2" target="_self" title="CLVECT_1:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E10:39"/>
<font color="Maroon" title="c5">z</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V3</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="clvect_1.html#E8:39"><i><font color="Green" title="E22">A12</font></i></a></i>;<br/>


</div>
