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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">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>thus </b><a NAME="E1:4"/>
( <font color="Maroon" title="c2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon" title="c2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font>
 ;<br/>

<a NAME="E2:4_1"/><b>then </b>
<font color="Maroon" title="c2">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 <span class="p1">(<span class="default"><a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font></span>)</span>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E3:4_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="rusub_1.html#D2" title="RUSUB_1:def.2">RUSUB_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:4_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


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

<b>thus </b><a NAME="E2:4"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">V</font> implies <font color="Maroon" title="c2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="rusub_1.html#K1" title="RUSUB_1:func.1">(0).</a> <font color="Maroon" title="c1">V</font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#T11" title="RUSUB_1:th.11">RUSUB_1:11</a></i>;<br/>


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