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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">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="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be  <a href="clvect_1.html#V1" title="CLVECT_1:attr.1">strict</a>  <a href="clvect_1.html#M1" title="CLVECT_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>thus </b><a NAME="E1:103"/>
( <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:103_1"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">W2</font>
 ;<br/>

<a NAME="E2:103_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:103_1_1"/><i><font color="Green" title="E66">A2</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></i>;<br/>
<a NAME="E2:103_1_1"/><i><font color="Green" title="E67">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 
<b>by </b><i><a class="ref" href="clvect_1.html#D5" target="_self" title="CLVECT_1:def.5">Def5</a></i>;<br/>
<b>thus </b><a NAME="E3:103_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="103_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:103_1_1_1"/><i><font color="Green" title="E68">A4</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E1:103_1_1"><i><font color="Green" title="E66">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font>;<br/>
<a NAME="E3:103_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:103_1_1_1"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E4:103_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:103_1"><i><font color="Green" title="E65">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">u</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="E6:103_1_1_1"/><i><font color="Green" title="E69">A5</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font>
 <b>and </b><br/><a NAME="E7:103_1_1_1"/><i><font color="Green" title="E70">A6</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 ;<br/>
<a NAME="E8:103_1_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E6:103_1_1_1"><i><font color="Green" title="E69">A5</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E9:103_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E7:103_1_1_1"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:103_1_1"/><i><font color="Green" title="E68">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="clvect_1.html#E2:103_1_1"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font>;<br/>
<a NAME="E6:103_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c4">W2</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E4:103_1_1"><i><font color="Green" title="E68">A7</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E7:103_1_1"/><b>then </b>
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W1</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E1:103_1"><i><font color="Green" title="E65">A1</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">u</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="E9:103_1_1"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c8">u</font>
 <b>and </b><br/><a NAME="E10:103_1_1"/><i><font color="Green" title="E70">A9</font></i>: 
<font color="Maroon" title="c8">u</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">W1</font>
 ;<br/>
<a NAME="E11:103_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">u</font>
 
<b>by </b><i><a class="txt" href="clvect_1.html#E9:103_1_1"><i><font color="Green" title="E69">A8</font></i></a>, <a class="ref" href="rlvect_1.html#T21" title="RLVECT_1:th.21">RLVECT_1:21</a></i>;<br/>
<b>hence </b><a NAME="E12:103_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">W1</font>
 <b>by </b><i><a class="txt" href="clvect_1.html#E10:103_1_1"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:103_1"/>
<font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font>
 <b>by </b><i><a class="ref" href="clvect_1.html#T50" target="_self" title="CLVECT_1:th.50">Th50</a></i>;<br/>


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

<b>thus </b><a NAME="E2:103"/>
( <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> implies <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> <a href="clvect_1.html#K5" title="CLVECT_1:func.5">+</a> <font color="Maroon" title="c4">W2</font> )
 ;<br/>


</div>
