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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Fy</font> being  <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">Fy</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/> <a href="card_fin.html#K3" title="CARD_FIN:func.3">Card_Intersection</a> <font color="Olive" title="b1">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">Fy</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Fy</font> be   <a href="finset_1.html#V2" title="FINSET_1:attr.2">finite-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  &amp; <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="card_fin.html#K3" title="CARD_FIN:func.3">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="card_fin.html#K3" title="CARD_FIN:func.3">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">Fy</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E39">A1</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">Ch</font> =  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c5">P</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="card_fin.html#E53"><span class="lab"><font color="Green" title="E37">Lm3</font></span></a></span>;<br/>
<a NAME="E6:59"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c1">k</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="card_fin.html#T11" target="_self" title="CARD_FIN:th.11">Th11</a></span>;<br/>
<a NAME="E7:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">XFS</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:59"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">P</font>
 <span class="kw">and </span><br/><a NAME="E10:59"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c6">XFS</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> holds <br/><font color="Maroon" title="c6">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Olive" title="b2">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:59"/><span class="lab"><font color="Green" title="E46">A8</font></span>: 
 <a href="card_fin.html#K3" title="CARD_FIN:func.3">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c6">XFS</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="card_fin.html#D4" target="_self" title="CARD_FIN:def.4">Def4</a></span>;<br/>
<a NAME="E12:59"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E6:59"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:59"/><span class="kw">then </span>
<font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/>
<a NAME="E14:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="afinsq_2.html#K6" title="AFINSQ_2:func.6">"**"</a> <font color="Maroon" title="c6">XFS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_2.html#T49" title="AFINSQ_2:th.49">AFINSQ_2:49</a></span>;<br/>
<a NAME="E15:59"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T87" title="CARD_1:th.87">CARD_1:87</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E16:59"/><span class="kw">then </span>
<font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">P0</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c3">X</font>,<span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span><span class="kw"> such that </span><br/><a NAME="E18:59"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<font color="Maroon" title="c7">P0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E19:59"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P0</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></span>;<br/>
<a NAME="E20:59"/>
<font color="Maroon" title="c7">P0</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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="#E19:59"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>;<br/>
<a NAME="E21:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A13</font></span>: 
 <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c7">P0</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#T36" target="_self" title="CARD_FIN:th.36">Th36</a></span>;<br/>
<a NAME="E22:59"/>
<font color="Maroon" title="c6">XFS</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c7">P0</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="txt" href="#E15:59"><span class="lab"><font color="Green" title="E48">A10</font></span></a>, <a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E49">A11</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:59"/>
 <a href="card_fin.html#K3" title="CARD_FIN:func.3">Card_Intersection</a> <font color="Maroon" title="c2">Fy</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">Fy</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E21:59"><span class="lab"><font color="Green" title="E51">A13</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="afinsq_2.html#T63" title="AFINSQ_2:th.63">AFINSQ_2:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
