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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">W1</font>, <font color="Maroon" title="c3">W2</font> be    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">W1</font> is    <a href="rusub_1.html#M1" title="RUSUB_1:mode.1">Subspace</a> of <font color="Maroon" title="c3">W2</font>
 ;<br/>

<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="E2:48"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c2">W1</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">u</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:48"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c2">W1</font>
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</font> <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">+</a> <font color="Maroon" title="c6">u</font>
 <b>by </b><i><a class="ref" href="rusub_2.html#T63" title="RUSUB_2:th.63">RUSUB_2:63</a></i>;<br/>
<a NAME="E6:48"/>
<font color="Maroon" title="c6">u</font> <a href="rlvect_1.html#R1" title="RLVECT_1:pred.1">in</a> <font color="Maroon" title="c3">W2</font>
 
<b>by </b><i><a class="txt" href="rusub_3.html#E1:48"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="rusub_3.html#E4:48"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="rusub_1.html#T1" title="RUSUB_1:th.1">RUSUB_1:1</a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">v</font> <a href="rusub_1.html#K3" title="RUSUB_1:func.3">+</a> <font color="Maroon" title="c3">W2</font>
 <b>by </b><i><a class="txt" href="rusub_3.html#E5:48"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="rusub_2.html#T63" title="RUSUB_2:th.63">RUSUB_2:63</a></i>;<br/>


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