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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#NK1" title="NUMBERS:NK.1">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><br/>  for <font color="Olive" title="b5">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   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="Olive" title="b5">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b3">J</font>,<font color="Olive" title="b4">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="c2">f</font>, <font color="Maroon" title="c3">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#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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><br/>  for <font color="Olive" title="b3">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   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="Olive" title="b3">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="c2">f</font>,<font color="Maroon" title="c3">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">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="c4">J</font>, <font color="Maroon" title="c5">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c4">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="b1">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="c2">f</font>,<font color="Maroon" title="c3">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c5">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">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="c6">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="c2">f</font>,<font color="Maroon" title="c3">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c5">K</font></span>)</span></span>)</span> )</span><br/>











<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">I</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#NK1" title="NUMBERS:NK.1">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#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b6">K</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b5">J</font>,<font color="Olive" title="b6">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="E1:38"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="E1:38_1"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:38_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">k</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:38_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/>
<a NAME="E2:38_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="numbers.html#K5" title="NUMBERS:func.5">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" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">I</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">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#NK1" title="NUMBERS:NK.1">NAT</a><br/>  for <font color="Olive" title="b4">J</font>, <font color="Olive" title="b5">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="b4">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="b1">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#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b5">K</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="b2">f</font>,<font color="Olive" title="b3">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b4">J</font>,<font color="Olive" title="b5">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Olive" title="b5">K</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font>, <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#NK1" title="NUMBERS:NK.1">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#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b4">K</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b3">J</font>,<font color="Olive" title="b4">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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#NK1" title="NUMBERS:NK.1">NAT</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">K</font> holds <br/> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Olive" title="b1">J</font>,<font color="Olive" title="b2">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font> implies  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="E3:38_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font> or  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="E4:38_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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="38_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">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="#E2:38_1_1"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:38_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="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>;<br/><a NAME="E2:38_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">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="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T11" onmouseover="rs('uproots/T11')" onmouseout="rh()">UPROOTS:11</a></span>;<br/><a NAME="E3:38_1_1_1_1"/><span class="lab"><font color="Green" title="E43">A8</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="#E4:38_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:38_1_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a></span>;<br/><a NAME="E4:38_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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="E5:38_1_1_1_1"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="#E3:38_1_1_1_1"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E2:38_1_1_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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">x</font> =  the    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ;<br/><a NAME="E2:38_1_1_1_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </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#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E1:38_1_1_1_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</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">object</a> <span class="kw"> such that </span><br/><a NAME="E4:38_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A10</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#T29" onmouseover="rs('card_1/T29')" onmouseout="rh()">CARD_1:29</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#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="E6:38_1_1_1_2"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="E7:38_1_1_1_2"/><span class="lab"><font color="Green" title="E44">A12</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#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#D3" onmouseover="rs('uproots/D3')" onmouseout="rh()">UPROOTS:def 3</a></span>;<br/><a NAME="E8:38_1_1_1_2"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="#E3:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E4:38_1_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T29" onmouseover="rs('zfmisc_1/T29')" onmouseout="rh()">ZFMISC_1:29</a></span>;<br/><a NAME="E9:38_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A14</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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E10:38_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="E11:38_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">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>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E12:38_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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#K24" title="BINOP_2:func.24">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="#E8:38_1_1_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E13:38_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a></span>;<br/><a NAME="E14:38_1_1_1_2"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="38_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#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_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="#E3:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E4:38_1_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">n1</font> = <font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k9</font>;<br/>
<span class="lab"><font color="Green" title="E49">A17</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span> = 
<a href="binop_2.html#K24" title="BINOP_2:func.24">multnat</a> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_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="funct_1.html#K1" title="FUNCT_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" onmouseover="rs('binop_2/D24')" onmouseout="rh()">BINOP_2:def 24</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">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" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">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#T75" onmouseover="rs('funct_3/T75')" onmouseout="rh()">FUNCT_3:75</a></span>
<br/>.= 
<a href="binop_2.html#K24" title="BINOP_2:func.24">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" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="funct_1.html#K1" title="FUNCT_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="#E11:38_1_1_1_2"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="#E9:38_1_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:38_1_1_1_2_1_2"/>
(  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="#E8:38_1_1_1_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T33" onmouseover="rs('zfmisc_1/T33')" onmouseout="rh()">ZFMISC_1:33</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E50">A18</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_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="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:38_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_1_1_2_1_2_1"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_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="#E6:38_1_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E7:38_1_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E1:38_1_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="rvsum_1.html#T72" onmouseover="rs('rvsum_1/T72')" onmouseout="rh()">RVSUM_1:72</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_2_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_2_1_2_2"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k9</font></span>)</span></span><br/><div class="add"><a NAME="E2:38_1_1_1_2_1_2_2"/><span class="kw">then </span>
 <a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="finseq_1.html#T94" onmouseover="rs('finseq_1/T94')" onmouseout="rh()">FINSEQ_1:94</a></span>;<br/><a NAME="E3:38_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="finseq_1.html#K9" title="FINSEQ_1:func.9">&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#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_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="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38_1_1_1_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E9:38_1_1_1_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E13:38_1_1_1_2"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="finseq_2.html#T34" onmouseover="rs('finseq_2/T34')" onmouseout="rh()">FINSEQ_2:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:38_1_1_1_2_1_2_2"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_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="#E6:38_1_1_1_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E3:38_1_1_1_2_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a>, <a class="ref" href="rvsum_1.html#T73" onmouseover="rs('rvsum_1/T73')" onmouseout="rh()">RVSUM_1:73</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E15:38_1_1_1_2"/>
 <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="#E4:38_1_1_1_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="nat_5.html#T27" target="_self" onmouseover="rs('nat_5/T27')" onmouseout="rh()">Th27</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:38_1_1_1_3"/><span class="lab"><font color="Green" title="E41">A19</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="E2:38_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="#E4:38_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">object</a> <span class="kw"> such that </span><br/><a NAME="E4:38_1_1_1_3"/><span class="lab"><font color="Green" title="E42">A20</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" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">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="#E4:38_1_1_1_3"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E6:38_1_1_1_3"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">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="#E1:38_1_1_1_3"><span class="lab"><font color="Green" title="E41">A19</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/><a NAME="E7:38_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A21</font></span>: 
( <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">k</font> &amp;  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="#E1:38_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" onmouseover="rs('xreal_1/T8')" onmouseout="rh()">XREAL_1:8</a></span>;<br/><a NAME="E8:38_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" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/><a NAME="E9:38_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A22</font></span>: 
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</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#K8" title="MCART_1:func.8">:]</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T104" onmouseover="rs('zfmisc_1/T104')" onmouseout="rh()">ZFMISC_1:104</a></span>;<br/><a NAME="E10:38_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" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E11:38_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" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E12:38_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A23</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">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#K4" title="CARD_1:func.4">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="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T41" onmouseover="rs('card_2/T41')" onmouseout="rh()">CARD_2:41</a></span>;<br/><span class="lab"><font color="Green" title="E46">A24</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" onmouseover="rs('xboole_1/T39')" onmouseout="rh()">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="#E4:38_1_1_1_3"><span class="lab"><font color="Green" title="E42">A20</font></span></a>, <a class="ref" href="zfmisc_1.html#T40" onmouseover="rs('zfmisc_1/T40')" onmouseout="rh()">ZFMISC_1:40</a></span>
;<br/><a NAME="E14:38_1_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</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#K8" title="MCART_1:func.8">:]</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="mcart_1.html#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T97" onmouseover="rs('zfmisc_1/T97')" onmouseout="rh()">ZFMISC_1:97</a></span>;<br/><span class="kw">hence </span> <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c13">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</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#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">J</font>,<font color="Maroon" title="c17">K0</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38_1_1_1_3"><span class="lab"><font color="Green" title="E44">A22</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" onmouseover="rs('nat_5/T26')" onmouseout="rh()">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="#E1:38_1"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E3:38_1_1"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E4:38_1_1"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E12:38_1_1_1_3"><span class="lab"><font color="Green" title="E45">A23</font></span></a>, <a class="txt" href="#E13:38_1_1_1_3"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="txt" href="#E7:38_1_1_1_3"><span class="lab"><font color="Green" title="E43">A21</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">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="#E13:38_1_1_1_3"><span class="lab"><font color="Green" title="E46">A24</font></span></a>, <a class="ref" href="nat_5.html#T26" target="_self" onmouseover="rs('nat_5/T26')" onmouseout="rh()">Th26</a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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


</div><span class="kw">end;</span></div>
<a NAME="E2:38"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">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" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E36">A1</font></span></a>);<br/></span>
<a NAME="E3:38"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">K</font>]
 
;<br/>
<span class="kw">hence </span><a NAME="E4:38"/>
( <font color="Maroon" title="c4">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="c6">j</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  <a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binop_2.html#K24" title="BINOP_2:func.24">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="c2">f</font>,<font color="Maroon" title="c3">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#K8" title="MCART_1:func.8">[:</a><span class="default"><font color="Maroon" title="c4">J</font>,<font color="Maroon" title="c5">K</font></span><a href="mcart_1.html#K8" title="MCART_1:func.8">:]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K2" title="UPROOTS:func.2">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="nat_5.html#K1" title="NAT_5:func.1">|</a> <font color="Maroon" title="c5">K</font></span>)</span></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
