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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="E1:3"/><i><font color="Green" title="E4">A1</font></i>: 
( the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</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="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</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="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</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">Y1</font>, <font color="Maroon" title="c3">Y2</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E2:3"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c2">Y1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">Y2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">Z</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E3:3"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Y2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">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="E5:3"/><i><font color="Green" title="E7">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="c5">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="E6:3"/><i><font color="Green" title="E8">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">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="E7:3"/><b>then </b><i><font color="Green" title="E9">A6</font></i>: 
 <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">Y1</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">Y1</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="c5">I</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bhsp_6.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="bhsp_6.html#E5:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="bhsp_6.html#T1" target="_self" title="BHSP_6:th.1">Th1</a></i>;<br/>
<a NAME="E8:3"/><i><font color="Green" title="E10">A7</font></i>: 
 <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c3">Y2</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="c3">Y2</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="c5">I</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="txt" href="bhsp_6.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="bhsp_6.html#E5:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="bhsp_6.html#E6:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="bhsp_6.html#T1" target="_self" title="BHSP_6:th.1">Th1</a></i>;<br/>
<a NAME="E9:3"/>
 <a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c4">Z</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="c5">I</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c2">Y1</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="c5">I</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K5" title="BHSP_5:func.5">setopfunc</a> <font color="Maroon" title="c3">Y2</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="c5">I</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="bhsp_6.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="bhsp_6.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="bhsp_6.html#E3:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="bhsp_6.html#E6:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="bhsp_5.html#T14" title="BHSP_5:th.14">BHSP_5:14</a></i>;<br/>
<b>hence </b><a NAME="E10:3"/>
 <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">Y1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c3">Y2</font></span>)</span>
 <b>by </b><i><a class="txt" href="bhsp_6.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="bhsp_6.html#E5:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="bhsp_6.html#E6:3"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="bhsp_6.html#E7:3"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="bhsp_6.html#E8:3"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="bhsp_6.html#T1" target="_self" title="BHSP_6:th.1">Th1</a></i>;<br/>


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