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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">j</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>, <font color="Olive" title="b2">g</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="b3">J</font>, <font color="Olive" title="b4">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="b3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b3">J</font>,<font color="Olive" title="b4">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="b2">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b4">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font>, <font color="Maroon" title="c4">g</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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c4">g</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="c5">J</font>, <font color="Maroon" title="c6">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="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c4">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c6">K</font></span>)</span></span>)</span> )</span><br/>











<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span>for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">f</font>, <font color="Olive" title="b4">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">I</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b5">J</font>, <font color="Olive" title="b6">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="Olive" title="b1">I</font>  st <font color="Olive" title="b5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b2">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b6">K</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b3">f</font>,<font color="Olive" title="b4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b5">J</font>,<font color="Olive" title="b6">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="b4">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b6">K</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:39"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:39_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</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">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:39_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:39_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c7">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/>
<a NAME="E2:39_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A3</font></span>: 
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">I</font>, <font color="Maroon" title="c9">j</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>, <font color="Olive" title="b2">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">I</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b3">J</font>, <font color="Olive" title="b4">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="c8">I</font>  st <font color="Olive" title="b3">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b4">K</font> holds <br/> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b3">J</font>,<font color="Olive" title="b4">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="b2">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b4">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c10">f</font>, <font color="Maroon" title="c11">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c8">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="c8">I</font>  st <font color="Olive" title="b1">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</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="c12">J</font>, <font color="Maroon" title="c13">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="c8">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="c12">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">K</font> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E3:39_1_1"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<font color="Maroon" title="c12">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">K</font> or  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:39_1_1"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_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="E1:39_1_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_1_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c14">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="E42">A7</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="E3:39_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A8</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="E4:39_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c13">K</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="#E4:39_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:39_1_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>;<br/><a NAME="E5:39_1_1_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><span class="kw">hence </span><a NAME="E6:39_1_1_1_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1_1"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E2:39_1_1_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E3:39_1_1_1_1"><span class="lab"><font color="Green" title="E43">A8</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="39_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A10</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E3:39_1_1_1_2"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:39_1_1_1_2"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">k9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:39_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
<font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T49" title="CARD_1:th.49">CARD_1:49</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">b</font> = <span class="p1">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">f9</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E7:39_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c17">f9</font>
 <span class="kw">and </span><br/><a NAME="E8:39_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<font color="Maroon" title="c17">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" title="UPROOTS:def.3">UPROOTS:def 3</a></span>;<br/><a NAME="E9:39_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:39_1_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T35" title="ZFMISC_1:th.35">ZFMISC_1:35</a></span>;<br/><a NAME="E10:39_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E11:39_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c8">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><a NAME="E12:39_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E13:39_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E14:39_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A17</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></span>;<br/><a NAME="E15:39_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A18</font></span>: 
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T37" title="PRE_POLY:th.37">PRE_POLY:37</a></span>;<br/><a NAME="E16:39_1_1_1_2"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c18">I9</font> = <font color="Maroon" title="c8">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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">j99</font> = <font color="Maroon" title="c9">j</font>, <font color="Maroon" title="c20">k99</font> = <font color="Maroon" title="c15">k9</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E5:39_1_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">n1</font> = <font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">n2</font> = <font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font>;<br/>
<span class="lab"><font color="Green" title="E50">A19</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span> = 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_2.html#D24" title="BINOP_2:def.24">BINOP_2:def 24</a></span>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c19">j99</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c20">k99</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" title="BINOP_1:def.1">BINOP_1:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">j99</font>,<font color="Maroon" title="c20">k99</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#T96" title="FUNCT_3:th.96">FUNCT_3:96</a></span>
<br/>.= 
<a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" title="BINOP_1:def.1">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39_1_1_1_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39_1_1_1_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2_1_2"><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="E1:39_1_1_1_2_1_2"/>
(  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E15:39_1_1_1_2"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E51">A20</font></span>: 
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span></span><br/><div class="add"><a NAME="E2:39_1_1_1_2_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></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="txt" href="#E3:39_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:39_1_1_1_2_1_2_1"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E8:39_1_1_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E1:39_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="rvsum_1.html#T102" title="RVSUM_1:th.102">RVSUM_1:102</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="39_1_1_1_2_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_2_1_2_2"/>
 <a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span></span><br/><div class="add"><a NAME="E2:39_1_1_1_2_1_2_2"/><span class="kw">then </span>
 <a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K14" title="PRE_POLY:func.14">support</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T6" title="UPROOTS:th.6">UPROOTS:6</a></span>;<br/><a NAME="E3:39_1_1_1_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">j</font>,<font color="Maroon" title="c15">k9</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39_1_1_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="txt" href="#E10:39_1_1_1_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E14:39_1_1_1_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="ref" href="bagorder.html#T3" title="BAGORDER:th.3">BAGORDER:3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_1_1_1_2_1_2_2"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_1_1_1_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E3:39_1_1_1_2_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="rvsum_1.html#T103" title="RVSUM_1:th.103">RVSUM_1:103</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>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E17:39_1_1_1_2"/>
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_1_1_1_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="nat_5.html#T27" target="_self" title="NAT_5:th.27">Th27</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="39_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:39_1_1_1_3"/><span class="lab"><font color="Green" title="E41">A21</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:39_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">K</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="#E4:39_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">k9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:39_1_1_1_3"/><span class="lab"><font color="Green" title="E42">A22</font></span>: 
<font color="Maroon" title="c14">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">K0</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">K1</font> = <font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">K0</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <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="c8">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></span>;<br/><a NAME="E6:39_1_1_1_3"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c17">K0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1_1_1_3"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="card_1.html#T50" title="CARD_1:th.50">CARD_1:50</a></span>;<br/><a NAME="E7:39_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A23</font></span>: 
( <span class="p1">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c17">K0</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></span>;<br/><a NAME="E8:39_1_1_1_3"/>
<font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c17">K0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>;<br/><a NAME="E9:39_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A24</font></span>: 
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T127" title="ZFMISC_1:th.127">ZFMISC_1:127</a></span>;<br/><a NAME="E10:39_1_1_1_3"/>
<font color="Maroon" title="c14">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">K0</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E11:39_1_1_1_3"/><span class="kw">then </span>
not <font color="Maroon" title="c14">k9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:39_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A25</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">K0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T54" title="CARD_2:th.54">CARD_2:54</a></span>;<br/><span class="lab"><font color="Green" title="E46">A26</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c17">K0</font> = 
<font color="Maroon" title="c13">K</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T39" title="XBOOLE_1:th.39">XBOOLE_1:39</a></span>
<br/>.= 
<font color="Maroon" title="c13">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_1_1_1_3"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="ref" href="zfmisc_1.html#T46" title="ZFMISC_1:th.46">ZFMISC_1:46</a></span>
;<br/><a NAME="E14:39_1_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T120" title="ZFMISC_1:th.120">ZFMISC_1:120</a></span>;<br/><span class="kw">hence </span> <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></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"><span class="p3">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c10">f</font>,<font color="Maroon" title="c11">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p3"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_1_1_1_3"><span class="lab"><font color="Green" title="E44">A24</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c17">K0</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:39_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E4:39_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E12:39_1_1_1_3"><span class="lab"><font color="Green" title="E45">A25</font></span></a>, <a class="txt" href="#E13:39_1_1_1_3"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="txt" href="#E7:39_1_1_1_3"><span class="lab"><font color="Green" title="E43">A23</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">k9</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c17">K0</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c11">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:39_1_1_1_3"><span class="lab"><font color="Green" title="E46">A26</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" title="NAT_5:th.26">Th26</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></span>
;<br/> <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>

</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E2:39"/>
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E1:39"><span class="lab"><font color="Green" title="E36">A1</font></span></a>);<br/></span>
<a NAME="E3:39"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c6">K</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E4:39"/>
( <font color="Maroon" title="c5">J</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K48" title="BINOP_2:func.48">multnat</a>  <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c4">g</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c5">J</font>,<font color="Maroon" title="c6">K</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c2">j</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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="c4">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c6">K</font></span>)</span></span>)</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>
