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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>;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green" title="E7">A1</font></i>: 
( the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  )
 ;<br/>

<b>let </b><font color="Maroon" title="c2">S</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="bhsp_5.html#M1" title="BHSP_5:mode.1">OrthogonalFamily</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green" title="E8">A2</font></i>: 
not <font color="Maroon" title="c2">S</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">H</font> be   <a href="hahnban.html#NM1" title="HAHNBAN:NM.1">Functional</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">H</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">S</font> holds <br/><font color="Maroon" title="c3">H</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> ) )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">I</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E5:6"/><i><font color="Green" title="E10">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c1">X</font> holds <br/><font color="Maroon" title="c4">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E6:6"/><i><font color="Green" title="E11">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">I</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="c1">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E7:6"/><b>then </b><i><font color="Green" title="E12">A6</font></i>: 
 <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c4">I</font>,the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E5:6"><i><font color="Green" title="E10">A4</font></i></a>, <a class="ref" href="bhsp_6.html#T1" title="BHSP_6:th.1">BHSP_6:1</a></i>;<br/>
<a NAME="E8:6"/>
( <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">I</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">S</font> holds <br/><font color="Maroon" title="c4">I</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) )
 
<b>by </b><i><a class="txt" href="bhsp_7.html#E6:6"><i><font color="Green" title="E11">A5</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">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" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="bhsp_1.html#K2" title="BHSP_1:func.2">.|.</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<font color="Maroon" title="c3">H</font>,<a href="binop_2.html#K33" title="BINOP_2:func.33">addreal</a> 
 <b>by </b><i><a class="txt" href="bhsp_7.html#E1:6"><i><font color="Green" title="E7">A1</font></i></a>, <a class="txt" href="bhsp_7.html#E2:6"><i><font color="Green" title="E8">A2</font></i></a>, <a class="txt" href="bhsp_7.html#E3:6"><i><font color="Green" title="E9">A3</font></i></a>, <a class="txt" href="bhsp_7.html#E7:6"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="bhsp_7.html#T2" target="_self" title="BHSP_7:th.2">Th2</a></i>;<br/>


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