<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <br/>  for <font color="Olive" title="b2">s</font>, <font color="Olive" title="b3">t</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b2">s</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">t</font> holds <br/> <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b3">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">s</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">t</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">s</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">t</font> holds <br/> <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">t</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">t</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">s</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">t</font> implies  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">s</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">t</font>)</span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E11">L08</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">t</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">s</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">t</font>)</span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">U</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">V</font> = <font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">t</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c1">a</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E3:17"/><span class="lab"><font color="Green" title="E12">A200</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E4:17"/><span class="lab"><font color="Green" title="E13">A290</font></span>: 
 for <font color="Olive" title="b1">W</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  holds   <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<font color="Olive" title="b1">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">W</font>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">W</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">W</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c8">W</font>)</span>)</span></span><br/>

<a NAME="E1:17_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">W</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c8">W</font></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E2:17_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A310</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">W</font>, <a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">FinS</a> (<font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">W</font>) <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D13" onmouseover="rs('rfunct_3/D13')" onmouseout="rh()">RFUNCT_3:def 13</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">fssu</font> = <font color="Maroon" title="c1">a</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c8">W</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">a</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T59" onmouseover="rs('relat_1/T59')" onmouseout="rh()">RELAT_1:59</a></span>;<br/>
<a NAME="E4:17_1"/><span class="lab"><font color="Green" title="E15">A320</font></span>: 
 <a href="glib_003.html#K1" title="GLIB_003:func.1">Seq</a> <font color="Maroon" title="c9">fssu</font>,<font color="Maroon" title="c9">fssu</font> <a href="classes1.html#R2" title="CLASSES1:pred.2">are_fiberwise_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dblseq_2.html#T51" onmouseover="rs('dblseq_2/T51')" onmouseout="rh()">DBLSEQ_2:51</a></span>;<br/>
<span class="kw">thus </span> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">W</font>) = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K20" title="RFUNCT_3:func.20">FinS</a> (<font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">W</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfunct_3.html#D14" onmouseover="rs('rfunct_3/D14')" onmouseout="rh()">RFUNCT_3:def 14</a></span>
<br/>.= 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c8">W</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E14">A310</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E15">A320</font></span></a>, <a class="ref" href="classes1.html#T76" onmouseover="rs('classes1/T76')" onmouseout="rh()">CLASSES1:76</a>, <a class="ref" href="rfinseq.html#T9" onmouseover="rs('rfinseq/T9')" onmouseout="rh()">RFINSEQ:9</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:17"/><span class="lab"><font color="Green" title="E14">A610</font></span>: 
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">s</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">s</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E13">A290</font></span></a></span>;<br/>
<a NAME="E6:17"/><span class="lab"><font color="Green" title="E15">A620</font></span>: 
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">t</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E13">A290</font></span></a></span>;<br/>
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">Sum</a> (<font color="Maroon" title="c7">F</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>) = 
 <a href="rvsum_1.html#K16" title="RVSUM_1:func.16">Sum</a> <span class="p1">(<span class="default"><a href="binpack1.html#K3" title="BINPACK1:func.3">Seq</a> (<font color="Maroon" title="c1">a</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">s</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c4">t</font></span>)</span></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E13">A290</font></span></a></span>
<br/>.= 
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">t</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T140" onmouseover="rs('relat_1/T140')" onmouseout="rh()">RELAT_1:140</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:17"/>
 <a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c4">t</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c3">s</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="binpack1.html#K4" title="BINPACK1:func.4">SumBin</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">t</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E12">A200</font></span></a>, <a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E11">L08</font></span></a>, <a class="ref" href="funct_1.html#T71" onmouseover="rs('funct_1/T71')" onmouseout="rh()">FUNCT_1:71</a>, <a class="ref" href="rfunct_3.html#T83" onmouseover="rs('rfunct_3/T83')" onmouseout="rh()">RFUNCT_3:83</a>, <a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E14">A610</font></span></a>, <a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E15">A620</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
