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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">J</font>, <font color="Olive" title="b3">K</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">J</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">K</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b2">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b3">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">J</font>, <font color="Olive" title="b2">K</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">J</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">K</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b1">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b2">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">J</font>, <font color="Maroon" title="c4">K</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">J</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">K</font> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:37"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">J</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">K</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon" title="c1">I</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not <font color="Maroon" title="c1">I</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><span class="kw">suppose </span></a><a NAME="E1:37_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c1">I</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c5">b</font> =  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><span class="lab"><font color="Green" title="E36">A3</font></span>:  <a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D13" title="PRE_POLY:def.13">PRE_POLY:def 13</a></span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/><a NAME="E3:37_1_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K17" title="PRE_POLY:func.17">EmptyBag</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></span>;<br/><a NAME="E4:37_1_1"/>
( <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:37_1_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:37_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E3:37_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><span class="kw">suppose </span></a><a NAME="E1:37_1_2"/>
not <font color="Maroon" title="c1">I</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">I9</font> = <font color="Maroon" title="c1">I</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><a NAME="E3:37_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">I9</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c5">I9</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T118" title="ZFMISC_1:th.118">ZFMISC_1:118</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c2">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c5">I9</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E5:37_1_2"/><span class="lab"><font color="Green" title="E35">A5</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="c6">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ;<br/><span class="kw">thus </span> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span> = 
 <a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c6">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">J</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T25" target="_self" title="NAT_5:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c3">J</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c4">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:37"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E5:37_1_2"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="ref" href="rfunct_3.html#T86" title="RFUNCT_3:th.86">RFUNCT_3:86</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K23" title="RFUNCT_3:func.23">Sum</a> <font color="Maroon" title="c6">F</font>,<font color="Maroon" title="c4">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T25" target="_self" title="NAT_5:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c3">J</font></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c4">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_5.html#T25" target="_self" title="NAT_5:th.25">Th25</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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