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

<span class="kw">let </span><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp; not <font color="Maroon" title="c2">Y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  implies  ex <font color="Olive" title="b1">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> <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"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
not <font color="Maroon" title="c1">X</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">and </span><br/><a NAME="E2:79"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
not <font color="Maroon" title="c2">Y</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>  st <br/>(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Olive" title="b1">F</font></span>)</span> <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"> <font color="Olive" title="b2">f</font> where <font color="Olive" title="b2">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b2">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  &amp; ( for <font color="Olive" title="b2">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span>;<br/>
<a NAME="E3:79"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_1"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<a NAME="E2:79_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="stirl2_1.html#T8" title="STIRL2_1:th.8">STIRL2_1:8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">n</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a></span>;<br/>
<a NAME="E4:79_1"/>
<font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E5:79_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">k</font> = <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></span>;<br/>
<a NAME="E7:79_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>,<font color="Maroon" title="c5">k</font> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span>]
 
;<br/>
<span class="kw">hence </span><a NAME="E8:79_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">XF</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> <font color="Maroon" title="c2">Y</font></span>)</span>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:79"/><span class="lab"><font color="Green" title="E58">A5</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:79"><span class="lab"><font color="Green" title="E57">A3</font></span></a>);<br/></span>
<span class="kw">set </span><font color="Maroon" title="c4">Onto</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : not $1 <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="Olive" title="b1">f</font> </span>}</span> ;<br/>
<a NAME="E6:79"/><span class="lab"><font color="Green" title="E59">A6</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_2"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span><br/>

<a NAME="E2:79_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_2_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span><br/>

<a NAME="E2:79_2_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp; not <font color="Maroon" title="c5">x</font> <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="Olive" title="b1">f</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:79_2_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_2"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:79_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">Fy9</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span><span class="kw"> such that </span><br/><a NAME="E8:79"/><span class="lab"><font color="Green" title="E60">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> holds <br/><font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S2" title="FUNCT_2:sch.2">FUNCT_2:sch 2</a>(<a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E59">A6</font></span></a>);<br/></span>
<a NAME="E9:79"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds  <font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_3_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Fy9</font> or  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Fy9</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_3_1_1"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Fy9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><div class="add"><a NAME="E2:79_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <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">Fy9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_3_1_1"/>
<font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_3_1_2"/>
not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Fy9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:79_3_1_2"/>
<font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">Fy</font> = <font color="Maroon" title="c5">Fy9</font> as   <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#D3" target="_self" title="CARD_FIN:def.3">Def3</a></span>;<br/>
<a NAME="E11:79"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">Fy9</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></span>;<br/>
<a NAME="E12:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A9</font></span>: 
 <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="c6">Fy</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">u</font> =  <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="c6">Fy</font></span>)</span> 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>  ;<br/>
<a NAME="E14:79"/><span class="lab"><font color="Green" title="E62">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">XF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">XF</font> = <font color="Maroon" title="c3">XF</font> as   <a href="afinsq_1.html#NM1" title="AFINSQ_1:NM.1">XFinSequence</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T7" title="AFINSQ_1:th.7">AFINSQ_1:7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">XF</font> = <font color="Maroon" title="c8">XF</font> as   <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>  ;<br/>
<a NAME="E17:79"/><span class="lab"><font color="Green" title="E63">A11</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="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="c9">XF</font> holds <br/> ex <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> &amp; ( for <font color="Olive" title="b4">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font> holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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"> ( <font color="Maroon" title="c10">n</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="c9">XF</font> implies  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_4"/><span class="lab"><font color="Green" title="E64">A12</font></span>: 
<font color="Maroon" title="c10">n</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="c9">XF</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b3">f</font>,<font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> ) )</span><br/>

<span class="kw">take </span>
 <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">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> &amp; ( 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="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Olive" title="b1">y</font> holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b2">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> ) )</span><br/>

<span class="kw">take </span>
1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; ( for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b1">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:79_4"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 holds <br/> <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="c6">Fy</font>,<font color="Olive" title="b1">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">f9</font> be   <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"> ( <font color="Maroon" title="c11">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 implies  <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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:79_4"/>
<font color="Maroon" title="c11">f9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K1" title="CARD_FIN:func.1">Choose</a> <font color="Maroon" title="c2">Y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">Y</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="E5:79_4"/><span class="lab"><font color="Green" title="E65">A13</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f9</font>
 <span class="kw">and </span><br/><a NAME="E6:79_4"/><span class="lab"><font color="Green" title="E66">A14</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="c12">f</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> <font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fin.html#D1" target="_self" title="CARD_FIN:def.1">Def1</a></span>;<br/>
<a NAME="E7:79_4"/><span class="lab"><font color="Green" title="E67">A15</font></span>: 
 <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_4_1"/><span class="lab"><font color="Green" title="E68">A16</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<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"> <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span></span><br/>

<a NAME="E2:79_4_1"/>
 <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="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_4"><span class="lab"><font color="Green" title="E66">A14</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_1.html#T142" title="FUNCT_1:th.142">FUNCT_1:142</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:79_4_1"/><span class="lab"><font color="Green" title="E69">A17</font></span>: 
<font color="Maroon" title="c14">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E5:79_4_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:79_4_1"/><span class="lab"><font color="Green" title="E70">A18</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_4_1"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="ref" href="card_fin.html#T24" target="_self" title="CARD_FIN:th.24">Th24</a></span>;<br/>
<a NAME="E7:79_4_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">x1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E4:79_4_1"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="txt" href="#E6:79_4_1"><span class="lab"><font color="Green" title="E70">A18</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E9:79_4_1"/><span class="lab"><font color="Green" title="E71">A19</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">and </span><br/><a NAME="E10:79_4_1"/>
not <font color="Maroon" title="c14">x1</font> <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="c15">g</font>
 ;<br/>
<a NAME="E11:79_4_1"/><span class="lab"><font color="Green" title="E72">A20</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c15">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">gy</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">gy</font> <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="c15">g</font> or <font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_4_1_1"/><span class="lab"><font color="Green" title="E73">A21</font></span>: 
<font color="Maroon" title="c16">gy</font> <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="c15">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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></span><br/>

<span class="kw">assume </span><a NAME="E2:79_4_1_1"/>
not <font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:79_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A22</font></span>: 
<font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_4_1_1"><span class="lab"><font color="Green" title="E73">A21</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:79_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E5:79_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A23</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">gy</font> <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="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E6:79_4_1_1"/>
<font color="Maroon" title="c16">gy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_4_1_1"><span class="lab"><font color="Green" title="E74">A22</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E7:79_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">gy</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_4_1"><span class="lab"><font color="Green" title="E68">A16</font></span></a>, <a class="txt" href="#E9:79_4_1"><span class="lab"><font color="Green" title="E71">A19</font></span></a>, <a class="txt" href="#E5:79_4_1_1"><span class="lab"><font color="Green" title="E75">A23</font></span></a>, <a class="ref" href="card_fin.html#D2" target="_self" title="CARD_FIN:def.2">Def2</a></span>;<br/>
<a NAME="E8:79_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c16">gy</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E1:79_4_1_1"><span class="lab"><font color="Green" title="E73">A21</font></span></a></span>;<br/>
<a NAME="E9:79_4_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c15">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">h</font> &amp; not <font color="Maroon" title="c16">gy</font> <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="Olive" title="b1">h</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E10:79_4_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_4_1_1"><span class="lab"><font color="Green" title="E73">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:79_4_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_4_1"><span class="lab"><font color="Green" title="E69">A17</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:79_4_1"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:79_4_1"><span class="lab"><font color="Green" title="E71">A19</font></span></a>, <a class="txt" href="#E11:79_4_1"><span class="lab"><font color="Green" title="E72">A20</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">I</font> =  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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>  ;<br/>
<a NAME="E9:79_4"/><span class="lab"><font color="Green" title="E68">A24</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="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_4"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>;<br/>
<a NAME="E10:79_4"/>
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">g9</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c14">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span> or <font color="Maroon" title="c14">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_4_2"/>
<font color="Maroon" title="c14">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c14">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:79_4_2"/><span class="lab"><font color="Green" title="E69">A25</font></span>: 
<font color="Maroon" title="c14">g9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">and </span><br/><a NAME="E4:79_4_2"/><span class="lab"><font color="Green" title="E70">A26</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E5:79_4_2"/><span class="lab"><font color="Green" title="E71">A27</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">gg</font> = <font color="Maroon" title="c15">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_4_2"><span class="lab"><font color="Green" title="E70">A26</font></span></a>, <a class="txt" href="#E5:79_4_2"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:79_4_2"/><span class="lab"><font color="Green" title="E72">A28</font></span>: 
<font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79_4"><span class="lab"><font color="Green" title="E66">A14</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E9:79_4_2"/>
not <font color="Maroon" title="c17">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_4_2"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="txt" href="#E8:79_4_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:79_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A29</font></span>: 
<font color="Maroon" title="c16">gg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">y</font>)
 
;<br/>
<a NAME="E11:79_4_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E12:79_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A30</font></span>: 
<font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font> <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">Fy9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79_4_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E13:79_4_2"/><span class="lab"><font color="Green" title="E75">A31</font></span>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:79_4_2_1"/><span class="lab"><font color="Green" title="E76">A32</font></span>: 
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E2:79_4_2_1"/><span class="lab"><font color="Green" title="E77">A33</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</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"> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font></span><br/>

<a NAME="E3:79_4_2_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_4_2_1"><span class="lab"><font color="Green" title="E77">A33</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:79_4_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">f</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_4_2_1"><span class="lab"><font color="Green" title="E76">A32</font></span></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></span>;<br/>
<a NAME="E5:79_4_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A34</font></span>: 
not <font color="Maroon" title="c18">z</font> <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="c16">gg</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_4_2"><span class="lab"><font color="Green" title="E71">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:79_4_2_1"/>
<font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c18">z</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E1:79_4_2_1"><span class="lab"><font color="Green" title="E76">A32</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:79_4_2_1"/>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_4_2_1"><span class="lab"><font color="Green" title="E78">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:79_4_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c17">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E8:79_4_2"><span class="lab"><font color="Green" title="E72">A28</font></span></a></span>;<br/>
<a NAME="E15:79_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">Fy</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_4_2"><span class="lab"><font color="Green" title="E74">A30</font></span></a>, <a class="txt" href="#E10:79_4_2"><span class="lab"><font color="Green" title="E73">A29</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:79_4_2"/>
<font color="Maroon" title="c14">g9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_4_2"><span class="lab"><font color="Green" title="E69">A25</font></span></a>, <a class="txt" href="#E13:79_4_2"><span class="lab"><font color="Green" title="E75">A31</font></span></a>, <a class="ref" href="card_fin.html#D2" target="_self" title="CARD_FIN:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:79_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A35</font></span>: 
 <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p3"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_fin.html#K2" title="CARD_FIN:func.2">Intersection</a> <font color="Maroon" title="c6">Fy</font>,<font color="Maroon" title="c12">f</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:79_4"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_4_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_4_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:79_4_3_1"/>
( <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_4_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:79_4_3_1_1"/>
<font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/><div class="add"><a NAME="E2:79_4_3_1_1"/><span class="kw">then </span>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c13">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></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="#E1:79"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E7:79_4"><span class="lab"><font color="Green" title="E67">A15</font></span></a>, <a class="txt" href="#E9:79_4"><span class="lab"><font color="Green" title="E68">A24</font></span></a>, <a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="newton.html#T16" title="NEWTON:th.16">NEWTON:16</a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_4_3_1_1"/>
 <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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E14:79"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E1:79_4"><span class="lab"><font color="Green" title="E64">A12</font></span></a>, <a class="txt" href="#E5:79_4"><span class="lab"><font color="Green" title="E65">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="79_4_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:79_4_3_1_2"/><span class="lab"><font color="Green" title="E70">A36</font></span>: 
<font color="Maroon" title="c2">Y</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span><br/><div class="add"><a NAME="E2:79_4_3_1_2"/>
<font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E14:79"><span class="lab"><font color="Green" title="E62">A10</font></span></a>, <a class="txt" href="#E1:79_4"><span class="lab"><font color="Green" title="E64">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:79_4_3_1_2"/>
 <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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79_4"><span class="lab"><font color="Green" title="E65">A13</font></span></a>, <a class="txt" href="#E11:79_4"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="txt" href="#E9:79_4"><span class="lab"><font color="Green" title="E68">A24</font></span></a>, <a class="txt" href="#E1:79_4_3_1_2"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="card_fin.html#T4" target="_self" title="CARD_FIN:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:79_4"/>
 <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="c6">Fy</font>,<font color="Maroon" title="c11">f9</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E18:79"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c9">XF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Fy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">F</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <span class="kw"> such that </span><br/><a NAME="E20:79"/><span class="lab"><font color="Green" title="E64">A37</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E21:79"/><span class="lab"><font color="Green" title="E65">A38</font></span>: 
 <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="c6">Fy</font></span>)</span></span>)</span> <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="c10">F</font>
 <span class="kw">and </span><br/><a NAME="E22:79"/><span class="lab"><font color="Green" title="E66">A39</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="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="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:79"><span class="lab"><font color="Green" title="E63">A11</font></span></a>, <a class="ref" href="card_fin.html#T69" target="_self" title="CARD_FIN:th.69">Th69</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c10">F</font></span>)</span> <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"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="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="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E23:79"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c10">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:79"><span class="lab"><font color="Green" title="E64">A37</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c10">F</font></span>)</span> <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"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="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="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> ) )</span><br/>

<a NAME="E24:79"/><span class="lab"><font color="Green" title="E67">A40</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c7">u</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c7">u</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79"><span class="lab"><font color="Green" title="E61">A9</font></span></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></span>;<br/>
<a NAME="E25:79"/><span class="lab"><font color="Green" title="E68">A41</font></span>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E3:79_5"/><span class="lab"><font color="Green" title="E69">A42</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E4:79_5"/><span class="lab"><font color="Green" title="E70">A43</font></span>: 
<font color="Maroon" title="c12">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ;<br/>
<span class="kw">assume </span><a NAME="E5:79_5"/><span class="lab"><font color="Green" title="E71">A44</font></span>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:79_5"/>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></span>;<br/>
<a NAME="E7:79_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">Fy</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_5"><span class="lab"><font color="Green" title="E69">A42</font></span></a>, <a class="txt" href="#E5:79_5"><span class="lab"><font color="Green" title="E71">A44</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Fyy</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:79_5"/><span class="lab"><font color="Green" title="E72">A45</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Fyy</font>
 <span class="kw">and </span><br/><a NAME="E10:79_5"/><span class="lab"><font color="Green" title="E73">A46</font></span>: 
<font color="Maroon" title="c13">Fyy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">Fy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E12:79_5"/><span class="lab"><font color="Green" title="E74">A47</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">Fy</font>
 <span class="kw">and </span><br/><a NAME="E13:79_5"/><span class="lab"><font color="Green" title="E75">A48</font></span>: 
<font color="Maroon" title="c6">Fy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">Fyy</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:79_5"><span class="lab"><font color="Green" title="E73">A46</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E14:79_5"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_5"><span class="lab"><font color="Green" title="E74">A47</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E15:79_5"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E9:79_5"><span class="lab"><font color="Green" title="E72">A45</font></span></a>, <a class="txt" href="#E13:79_5"><span class="lab"><font color="Green" title="E75">A48</font></span></a></span>;<br/>
<a NAME="E16:79_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A49</font></span>: 
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> st <br/>( <font color="Maroon" title="c12">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Olive" title="b1">g</font> &amp; not <font color="Maroon" title="c14">y</font> <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="Olive" title="b1">g</font> )
 
;<br/>
<a NAME="E17:79_5"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_5"><span class="lab"><font color="Green" title="E74">A47</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:79_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_5"><span class="lab"><font color="Green" title="E70">A43</font></span></a>, <a class="txt" href="#E16:79_5"><span class="lab"><font color="Green" title="E76">A49</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E26:79"/><span class="lab"><font color="Green" title="E69">A50</font></span>: 
<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span> or <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_6"/><span class="lab"><font color="Green" title="E70">A51</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</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="c6">Fy</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:79_6"/><span class="lab"><font color="Green" title="E71">A52</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E4:79_6"/><span class="lab"><font color="Green" title="E72">A53</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_6"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f</font> = <font color="Maroon" title="c12">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79_6"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:79_6"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:79_6"/><span class="kw">then </span>
not <font color="Maroon" title="c13">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_6"><span class="lab"><font color="Green" title="E71">A52</font></span></a></span>;<br/>
<a NAME="E8:79_6"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E9:79_6"/><span class="kw">then </span>
not <font color="Maroon" title="c2">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c13">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:79_6"/><span class="lab"><font color="Green" title="E73">A54</font></span>: 
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <span class="kw">and </span><br/><a NAME="E12:79_6"/><span class="lab"><font color="Green" title="E74">A55</font></span>: 
not <font color="Maroon" title="c14">y</font> <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="c13">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E13:79_6"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">Fy9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:79_6"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E14:79_6"/><span class="kw">then </span>
<font color="Maroon" title="c5">Fy9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> <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">Fy9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E15:79_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A56</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">y</font>) <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">Fy9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:79"><span class="lab"><font color="Green" title="E60">A8</font></span></a>, <a class="txt" href="#E11:79_6"><span class="lab"><font color="Green" title="E73">A54</font></span></a></span>;<br/>
<a NAME="E16:79_6"/>
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c14">y</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:79_6"><span class="lab"><font color="Green" title="E74">A55</font></span></a></span>;<br/>
<a NAME="E17:79_6"/><span class="kw">then </span>
<font color="Maroon" title="c13">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c6">Fy</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:79_6"><span class="lab"><font color="Green" title="E75">A56</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:79_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_6"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="txt" href="#E3:79_6"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E27:79"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="card_fin.html#T4" target="_self" title="CARD_FIN:th.4">Th4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:79"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">f</font> where <font color="Olive" title="b1">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font> : <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <span class="p1">(<span class="default"><a href="afinsq_2.html#K7" title="AFINSQ_2:func.7">Sum</a> <font color="Maroon" title="c10">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:79"><span class="lab"><font color="Green" title="E65">A38</font></span></a>, <a class="txt" href="#E26:79"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E25:79"><span class="lab"><font color="Green" title="E68">A41</font></span></a>, <a class="txt" href="#E24:79"><span class="lab"><font color="Green" title="E67">A40</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="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="c10">F</font> holds <br/><font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">n</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"> ( <font color="Maroon" title="c11">n</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="c10">F</font> implies <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E29:79"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
<font color="Maroon" title="c11">n</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="c10">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><br/>

<a NAME="E30:79"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:79"><span class="lab"><font color="Green" title="E66">A39</font></span></a>, <a class="txt" href="#E29:79"><span class="lab"><font color="Green" title="E70">A57</font></span></a></span>;<br/>
<a NAME="E31:79"/>
<font color="Maroon" title="c9">XF</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:79"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="txt" href="#E20:79"><span class="lab"><font color="Green" title="E64">A37</font></span></a>, <a class="txt" href="#E29:79"><span class="lab"><font color="Green" title="E70">A57</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E32:79"/>
<font color="Maroon" title="c10">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binop_2.html#K19" title="BINOP_2:func.19">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="newton.html#K8" title="NEWTON:func.8">choose</a> <span class="p3">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K22" title="BINOP_2:func.22">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">Y</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> <font color="Maroon" title="c11">n</font></span>)</span> <a href="binop_2.html#K21" title="BINOP_2:func.21">-</a> 1</span>)</span> <a href="newton.html#K2" title="NEWTON:func.2">|^</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:79"><span class="lab"><font color="Green" title="E71">A58</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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