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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being   <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b2">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b2">Z</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">Y</font> holds <br/><font color="Olive" title="b2">Z</font> is    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c2">Y</font> be    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Z</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Z</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">Z</font> is    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font></span><br/><b>let </b><font color="Maroon" title="c3">Z</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">Z</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font> implies <font color="Maroon" title="c3">Z</font> is    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font> )</span><br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">Z</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">Z</font> is    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:8"/>
( <font color="Maroon" title="c2">Y</font> is    <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font> &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 
<b>by </b><i><a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>;<br/>
<a NAME="E3:8"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c3">Z</font> is    <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="bhsp_7.html#T4" target="_self" title="BHSP_7:th.4">Th4</a></i>;<br/>
<a NAME="E4:8"/>
for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Z</font> holds <br/><font color="Olive" title="b1">x</font> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:8"/>
<font color="Maroon" title="c3">Z</font> is    <a href="bhsp_5.html#M2" title="BHSP_5:mode.2">OrthonormalFamily</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="bhsp_7.html#E3:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="bhsp_5.html#D9" title="BHSP_5:def.9">BHSP_5:def 9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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