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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="E1:2"/><i><font color="Green" title="E3">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">Y</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>let </b><font color="Maroon" title="c3">I</font> be   <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/>



<b>assume </b><a NAME="E2:2"/><i><font color="Green" title="E4">A2</font></i>: 
( <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">I</font> &amp; ( 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>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">I</font> holds <br/><font color="Maroon" title="c3">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> ) )
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">p</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E4:2"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c4">p</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">X</font> <a href="finsop_1.html#K2" title="FINSOP_1:func.2">"**"</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="bhsp_6.html#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="bhsp_6.html#D1" target="_self" title="BHSP_6:def.1">Def1</a></i>;<br/>
<a NAME="E5:2"/><i><font color="Green" title="E6">A4</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">xd</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:2_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
( <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> iff <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 <b>by </b><i><a class="ref" href="bhsp_5.html#D4" title="BHSP_5:def.4">BHSP_5:def 4</a></i>;<br/><a NAME="E2:2_1_1"/>
( <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:2_1_1"/>
( <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> iff <font color="Maroon" title="c5">xd</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> )
 <b>by </b><i><a class="txt" href="bhsp_6.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="bhsp_6.html#E4:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="bhsp_6.html#E1:2_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:2_1"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:2_2"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span>
 ;<br/><a NAME="E2:2_2"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">s</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="bhsp_6.html#E1:2_2"><i><font color="Green" title="E7">A6</font></i></a></i>;<br/><a NAME="E4:2_2"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="txt" href="bhsp_6.html#E5:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="bhsp_6.html#E1:2_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></i>;<br/><span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="ref" href="bhsp_5.html#D4" title="BHSP_5:def.4">BHSP_5:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c3">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_6.html#E5:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="bhsp_6.html#E1:2_2"><i><font color="Green" title="E7">A6</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="txt" href="bhsp_6.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="bhsp_6.html#E4:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="bhsp_6.html#E4:2_2"><i><font color="Green" title="E8">A7</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:2_2"/>
<span class="p1">(<span class="default"><a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E7:2"/><b>then </b>
 <a href="bhsp_5.html#K4" title="BHSP_5:func.4">Func_Seq</a> <font color="Maroon" title="c3">I</font>,<font color="Maroon" title="c4">p</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">p</font>
 
<b>by </b><i><a class="txt" href="bhsp_6.html#E5:2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E8:2"/>
 <a href="bhsp_6.html#K1" title="BHSP_6:func.1">setsum</a> <font color="Maroon" title="c2">Y</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">Y</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="c3">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#E1:2"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="bhsp_6.html#E2:2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="bhsp_6.html#E4:2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="bhsp_5.html#D5" title="BHSP_5:def.5">BHSP_5:def 5</a></i>;<br/>


</div>
