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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E38</font></i>: 
( the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> is <a href="binop_1.html#V2">associative</a> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  )
 ;<br/>

<b>let </b><font color="Maroon">S</font> be  <a href="finset_1.html#V1">finite</a>  <a href="bhsp_5.html#M1">OrthogonalFamily</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">S</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon">H</font> be   <a href="hahnban.html#NM1">Functional</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">H</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="bhsp_1.html#K2">.|.</a> <font color="Olive">x</font> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon">I</font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E5:6"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/><font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<font color="Maroon">I</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font>
 
<b>by </b><i>, , <a class="ref" href="bhsp_6.html#T1">BHSP_6:1</a></i>;<br/>
<a NAME="E8:6"/>
( <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> &amp; ( for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> holds <br/><font color="Maroon">I</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> ) )
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a>, </i>;<br/>
<b>hence </b><a NAME="E9:6"/>
<span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">S</font></span>)</span> <a href="bhsp_1.html#K2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1">setsum</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K5">setopfunc</a> <font color="Maroon">S</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,<a href="numbers.html#K1">REAL</a> ,<font color="Maroon">H</font>,<a href="binop_2.html#K33">addreal</a> 
 <b>by </b><i>, , , , </i>;<br/>


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