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

<span class="kw">let </span><font color="Maroon" title="c1">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="firebrick">::  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> holds   <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Olive" title="b1">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>), <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</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="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>), <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </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> ex <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; (  for <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">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="c2">f</font> holds <br/>( ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">y</font> ) ) ) );<br/>
<a NAME="E1:59"/><span class="lab"><font color="Green" title="E34">A1</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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) holds <br/> ex <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<span class="kw">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>  =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>
<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="firebrick">::  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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) implies  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">z</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/>
<font color="Maroon" title="c3">x</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> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">z</font>]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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="E3:59_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:59_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <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">deffunc </span><font color="Maroon">H<sub>2</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"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<a NAME="E6:59_1"/>
 ex <font color="Olive" title="b1">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; (  for <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="c2">f</font> holds <br/>( ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] implies <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">y</font>) ) &amp; (  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>] implies <font color="Olive" title="b1">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">y</font>) ) ) ) )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="partfun1.html#S1" title="PARTFUN1:sch.1">PARTFUN1:sch 1</a>();<br/></span>
<span class="kw">hence </span><a NAME="E7:59_1"/>
 ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">z</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c3">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="E3:59"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) &amp; (  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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" title="CLASSES1:sch.1">CLASSES1:sch 1</a>(<a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E34">A1</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E4:59"/>
<font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> or  not <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> or  not <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c3">F</font> or  not <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59_2"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c4">x</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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E2:59_2"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c5">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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E3:59_2"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/>

<a NAME="E4:59_2"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 ex <font color="Olive" title="b1">f2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f2</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E37">A5</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">consider </span><font color="Maroon" title="c6">g1</font>, <font color="Maroon" title="c7">h1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:59_2"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g1</font>
 <span class="kw">and </span><br/><a NAME="E7:59_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font>
 <span class="kw">and </span><br/><a NAME="E8:59_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E9:59_2"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c2">f</font> holds <br/>( ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">g2</font>, <font color="Maroon" title="c9">h2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E11:59_2"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font>
 <span class="kw">and </span><br/><a NAME="E12:59_2"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h2</font>
 <span class="kw">and </span><br/><a NAME="E13:59_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E14:59_2"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c2">f</font> holds <br/>( ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c8">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E2:59_2"><span class="lab"><font color="Green" title="E37">A5</font></span></a></span>;<br/>
<a NAME="E15:59_2"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
 ex <font color="Olive" title="b1">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E1:59_2"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> implies <font color="Maroon" title="c6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_2_1"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font></span><br/><a NAME="E2:59_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/><a NAME="E3:59_2_1"/><span class="lab"><font color="Green" title="E49">A17</font></span>: 
( <font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="c2">f</font> &amp; <font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/><a NAME="E4:59_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A18</font></span>: 
<font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c8">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59_2"><span class="lab"><font color="Green" title="E45">A13</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">g91</font> = <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>, <font color="Maroon" title="c12">g92</font> = <font color="Maroon" title="c9">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E11:59_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E12:59_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:59_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></span>;<br/><a NAME="E6:59_2_1"/><span class="lab"><font color="Green" title="E51">A19</font></span>: 
<font color="Maroon" title="c8">g2</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">g92</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E11:59_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:59_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E4:59_2_1"><span class="lab"><font color="Green" title="E50">A18</font></span></a>, <a class="ref" href="funct_5.html#T22" title="FUNCT_5:th.22">FUNCT_5:22</a></span>;<br/><a NAME="E7:59_2_1"/>
<font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E3:59_2_1"><span class="lab"><font color="Green" title="E49">A17</font></span></a></span>;<br/><a NAME="E8:59_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">g1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g91</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E6:59_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="txt" href="#E2:59_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="ref" href="funct_5.html#T22" title="FUNCT_5:th.22">FUNCT_5:22</a></span>;<br/><span class="kw">hence </span><a NAME="E9:59_2_1"/>
<font color="Maroon" title="c6">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">g2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E7:59_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E12:59_2"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E2:59_2_1"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E6:59_2_1"><span class="lab"><font color="Green" title="E51">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:59_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:59_2"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E4:59_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E6:59_2"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="txt" href="#E11:59_2"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E5:59"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E6:59"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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="c3">F</font> or <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3"/>
<font color="Maroon" title="c4">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="c3">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_3"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
<font color="Maroon" title="c5">x</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="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E4:59_3"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:59_3"/><span class="lab"><font color="Green" title="E38">A22</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E7:59_3"/><span class="lab"><font color="Green" title="E39">A23</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E8:59_3"/><span class="lab"><font color="Green" title="E40">A24</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E9:59_3"/><span class="lab"><font color="Green" title="E41">A25</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c2">f</font> holds <br/>( ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E3:59_3"><span class="lab"><font color="Green" title="E36">A20</font></span></a></span>;<br/>
<a NAME="E10:59_3"/><span class="lab"><font color="Green" title="E42">A26</font></span>: 
 ex <font color="Olive" title="b1">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E3:59_3"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E43">A27</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</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> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> implies <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_3_1"/>
<font color="Maroon" title="c8">z</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> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:59_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A28</font></span>: 
<font color="Maroon" title="c8">z</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="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/><a NAME="E3:59_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A29</font></span>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span>)</span>,<font color="Maroon" title="c1">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/><div><span class="lab"><font color="Green" title="E46">A30</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>;<br/><span class="kw">assume </span><a NAME="E1:59_3_1_1"/><span class="lab"><font color="Green" title="E47">A31</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:59_3_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/><a NAME="E3:59_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A32</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="txt" href="#E1:59_3_1_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">k</font> = <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E10:59_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></span>;<br/><a NAME="E5:59_3_1_1"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<font color="Maroon" title="c8">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E10:59_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E3:59_3_1_1"><span class="lab"><font color="Green" title="E48">A32</font></span></a>, <a class="ref" href="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></span>;<br/><a NAME="E6:59_3_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T34" title="FUNCT_5:th.34">FUNCT_5:34</a></span>;<br/><a NAME="E7:59_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A34</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E10:59_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E8:59_3_1_1"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3_1_1"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="ref" href="funct_5.html#T34" title="FUNCT_5:th.34">FUNCT_5:34</a></span>;<br/><a NAME="E9:59_3_1_1"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:59_3_1_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">k</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">k</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">b</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="c10">k</font> or <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_3_1_1_1_1"/>
<font color="Maroon" title="c11">b</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="c10">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_3_1_1_1_1"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E4:59_3_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_1_1_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:59_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A38</font></span>: 
<font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E6:59_3_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A39</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E7:59_3_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c12">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_1_1_1_1"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E8:59_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E10:59_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E6:59_3_1_1_1_1"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:59_3_1_1_1_1"/>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3_1_1_1_1"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c11">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> or <font color="Maroon" title="c11">b</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="c10">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:59_3_1_1_1"/><span class="lab"><font color="Green" title="E53">A40</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">b</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="c10">k</font></span><br/>

<a NAME="E3:59_3_1_1_1"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:59_3_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E5:59_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:59_3"><span class="lab"><font color="Green" title="E38">A22</font></span></a>, <a class="txt" href="#E10:59_3"><span class="lab"><font color="Green" title="E42">A26</font></span></a>, <a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="txt" href="#E2:59_3_1_1_1"><span class="lab"><font color="Green" title="E53">A40</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/>
<a NAME="E6:59_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>;<br/>
<a NAME="E7:59_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_1_1_1"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E8:59_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">b</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_3_1_1_1"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:59_3_1_1_1"/>
<font color="Maroon" title="c11">b</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="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3_1_1"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E10:59_3_1_1"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="txt" href="#E1:59_3_1_1"><span class="lab"><font color="Green" title="E47">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:59_3_1_1"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_3_1"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="txt" href="#E7:59_3_1_1"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="txt" href="#E9:59_3_1_1"><span class="lab"><font color="Green" title="E52">A36</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="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_3_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:59_3_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <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="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:59_3_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_3"><span class="lab"><font color="Green" title="E41">A25</font></span></a>, <a class="txt" href="#E2:59_3_1"><span class="lab"><font color="Green" title="E44">A28</font></span></a>, <a class="txt" href="#E3:59_3_1"><span class="lab"><font color="Green" title="E45">A29</font></span></a>, <a class="ref" href="funct_5.html#T57" title="FUNCT_5:th.57">FUNCT_5:57</a></span>;<br/><span class="kw">hence </span><a NAME="E3:59_3_1_2"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:59_3_1"/>
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_3_1"><span class="lab"><font color="Green" title="E46">A30</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:59_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_3"><span class="lab"><font color="Green" title="E40">A24</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:59_3"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_3"><span class="lab"><font color="Green" title="E37">A21</font></span></a>, <a class="txt" href="#E7:59_3"><span class="lab"><font color="Green" title="E39">A23</font></span></a>, <a class="txt" href="#E11:59_3"><span class="lab"><font color="Green" title="E43">A27</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:59"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">s</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:59"/><span class="lab"><font color="Green" title="E36">A43</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E10:59"/><span class="lab"><font color="Green" title="E37">A44</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:59"/><span class="lab"><font color="Green" title="E38">A45</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> holds <br/><font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<a NAME="E12:59"/><span class="lab"><font color="Green" title="E39">A46</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/>
<a NAME="E13:59"/><span class="lab"><font color="Green" title="E40">A47</font></span>: 
 <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D4" title="FUNCT_5:def.4">FUNCT_5:def 4</a></span>;<br/>
<a NAME="E14:59"/><span class="lab"><font color="Green" title="E41">A48</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:59_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> or <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_4_1"/><span class="lab"><font color="Green" title="E42">A49</font></span>: 
<font color="Maroon" title="c6">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z1</font>, <font color="Maroon" title="c8">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_4_1"/><span class="lab"><font color="Green" title="E43">A50</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">z1</font>,<font color="Maroon" title="c8">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="ref" href="funct_6.html#T2" target="_self" title="FUNCT_6:th.2">Th11</a></span>;<br/>
<a NAME="E4:59_4_1"/><span class="lab"><font color="Green" title="E44">A51</font></span>: 
( <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z1</font> &amp; <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_4_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E5:59_4_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z2</font>,<font color="Maroon" title="c7">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="txt" href="#E1:59_4_1"><span class="lab"><font color="Green" title="E42">A49</font></span></a>, <a class="txt" href="#E3:59_4_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="ref" href="funct_4.html#T42" title="FUNCT_4:th.42">FUNCT_4:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c10">u</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c11">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:59_4_1"/><span class="lab"><font color="Green" title="E45">A52</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z2</font>,<font color="Maroon" title="c7">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:59_4_1"/><span class="lab"><font color="Green" title="E46">A53</font></span>: 
<font color="Maroon" title="c9">a</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="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E9:59_4_1"/><span class="lab"><font color="Green" title="E47">A54</font></span>: 
<font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font>
 <span class="kw">and </span><br/><a NAME="E10:59_4_1"/><span class="lab"><font color="Green" title="E48">A55</font></span>: 
<font color="Maroon" title="c11">b</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="c10">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></span>;<br/>
<a NAME="E11:59_4_1"/><span class="lab"><font color="Green" title="E49">A56</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z2</font>,<font color="Maroon" title="c7">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E12:59_4_1"/>
( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c1">X</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E8:59_4_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E9:59_4_1"><span class="lab"><font color="Green" title="E47">A54</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/>
<a NAME="E13:59_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A57</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c10">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_4_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E11:59_4_1"><span class="lab"><font color="Green" title="E49">A56</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E14:59_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c11">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z2</font>,<font color="Maroon" title="c7">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:59_4_1"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E3:59_4_1"><span class="lab"><font color="Green" title="E43">A50</font></span></a>, <a class="txt" href="#E7:59_4_1"><span class="lab"><font color="Green" title="E45">A52</font></span></a>, <a class="txt" href="#E8:59_4_1"><span class="lab"><font color="Green" title="E46">A53</font></span></a>, <a class="txt" href="#E10:59_4_1"><span class="lab"><font color="Green" title="E48">A55</font></span></a>, <a class="txt" href="#E4:59_4_1"><span class="lab"><font color="Green" title="E44">A51</font></span></a>, <a class="txt" href="#E11:59_4_1"><span class="lab"><font color="Green" title="E49">A56</font></span></a>, <a class="txt" href="#E13:59_4_1"><span class="lab"><font color="Green" title="E50">A57</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span> or <font color="Maroon" title="c6">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:59_4"/><span class="lab"><font color="Green" title="E42">A58</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span></span><br/>

<a NAME="E3:59_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A59</font></span>: 
( <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/>
<a NAME="E4:59_4"/><span class="lab"><font color="Green" title="E44">A60</font></span>: 
<font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <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="c2">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_4"><span class="lab"><font color="Green" title="E42">A58</font></span></a>, <a class="ref" href="card_3.html#T22" title="CARD_3:th.22">CARD_3:22</a></span>;<br/>
<a NAME="E5:59_4"/><span class="kw">then </span>
( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span></span>)</span>,<font color="Maroon" title="c1">X</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/>
<a NAME="E6:59_4"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">j</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<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">hence </span><a NAME="E7:59_4"/>
<font color="Maroon" title="c6">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E4:59_4"><span class="lab"><font color="Green" title="E44">A60</font></span></a>, <a class="txt" href="#E3:59_4"><span class="lab"><font color="Green" title="E43">A59</font></span></a>, <a class="ref" href="funct_5.html#T39" title="FUNCT_5:th.39">FUNCT_5:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:59"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">d</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">d</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> or <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_5"/>
<font color="Maroon" title="c6">d</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:59_5"/><span class="lab"><font color="Green" title="E42">A61</font></span>: 
<font color="Maroon" title="c7">z</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> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:59_5"/><span class="lab"><font color="Green" title="E43">A62</font></span>: 
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:59_5"/><span class="lab"><font color="Green" title="E44">A63</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z1</font>,<font color="Maroon" title="c9">z2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="txt" href="#E3:59_5"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="ref" href="funct_6.html#T2" target="_self" title="FUNCT_6:th.2">Th11</a></span>;<br/>
<a NAME="E7:59_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z2</font>,<font color="Maroon" title="c8">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E40">A47</font></span></a>, <a class="txt" href="#E3:59_5"><span class="lab"><font color="Green" title="E42">A61</font></span></a>, <a class="txt" href="#E6:59_5"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="ref" href="funct_4.html#T42" title="FUNCT_4:th.42">FUNCT_4:42</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">u</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c12">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:59_5"/><span class="lab"><font color="Green" title="E45">A64</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z2</font>,<font color="Maroon" title="c8">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E10:59_5"/><span class="lab"><font color="Green" title="E46">A65</font></span>: 
( <font color="Maroon" title="c10">a</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="c5">s</font> &amp; <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )
 <span class="kw">and </span><br/><a NAME="E11:59_5"/><span class="lab"><font color="Green" title="E47">A66</font></span>: 
<font color="Maroon" title="c12">b</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="c11">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></span>;<br/>
<a NAME="E12:59_5"/><span class="lab"><font color="Green" title="E48">A67</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z2</font>,<font color="Maroon" title="c8">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E13:59_5"/>
( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<font color="Maroon" title="c1">X</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E10:59_5"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/>
<a NAME="E14:59_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A68</font></span>: 
 ex <font color="Olive" title="b1">j</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c11">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_5"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E12:59_5"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E15:59_5"/><span class="lab"><font color="Green" title="E50">A69</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">b</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">z2</font>,<font color="Maroon" title="c8">z1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E16:59_5"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c12">b</font>,<font color="Maroon" title="c10">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">u</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59_5"><span class="lab"><font color="Green" title="E46">A65</font></span></a>, <a class="txt" href="#E11:59_5"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="ref" href="funct_5.html#T39" title="FUNCT_5:th.39">FUNCT_5:39</a></span>;<br/>
<a NAME="E17:59_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">d</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="c11">u</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_5"><span class="lab"><font color="Green" title="E43">A62</font></span></a>, <a class="txt" href="#E6:59_5"><span class="lab"><font color="Green" title="E44">A63</font></span></a>, <a class="txt" href="#E9:59_5"><span class="lab"><font color="Green" title="E45">A64</font></span></a>, <a class="txt" href="#E11:59_5"><span class="lab"><font color="Green" title="E47">A66</font></span></a>, <a class="txt" href="#E12:59_5"><span class="lab"><font color="Green" title="E48">A67</font></span></a>, <a class="txt" href="#E15:59_5"><span class="lab"><font color="Green" title="E50">A69</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:59_5"/>
<font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59_5"><span class="lab"><font color="Green" title="E49">A68</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E16:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A70</font></span>: 
 <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</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="c3">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E41">A48</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">then </span><span class="kw">consider </span><font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E18:59"/><span class="lab"><font color="Green" title="E43">A71</font></span>: 
 <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font>
 <span class="kw">and </span><br/><a NAME="E19:59"/><span class="lab"><font color="Green" title="E44">A72</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font>
 <span class="kw">and </span><br/><a NAME="E20:59"/><span class="lab"><font color="Green" title="E45">A73</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font>
 <span class="kw">and </span><br/><a NAME="E21:59"/><span class="lab"><font color="Green" title="E46">A74</font></span>: 
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">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="c2">f</font> holds <br/>( ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; ( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</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="c2">f</font> implies <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_6"/><span class="lab"><font color="Green" title="E47">A75</font></span>: 
<font color="Maroon" title="c8">z</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="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:59_6"/><span class="kw">then </span>
( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K8" title="FUNCT_6:func.8">Funcs</a> (<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">X</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span>)</span>,<font color="Maroon" title="c1">X</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E37">A44</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E38">A45</font></span></a>, <a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="ref" href="funct_6.html#D8" target="_self" title="FUNCT_6:def.8">Def8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">v</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E4:59_6"/><span class="lab"><font color="Green" title="E48">A76</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font>
 <span class="kw">and </span><br/><a NAME="E5:59_6"/><span class="lab"><font color="Green" title="E49">A77</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">and </span><br/><a NAME="E6:59_6"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <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/><div><span class="lab"><font color="Green" title="E50">A78</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c10">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>;<br/><span class="kw">assume </span><a NAME="E1:59_6_1"/><span class="lab"><font color="Green" title="E51">A79</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:59_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A80</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"> the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E1:59_6"><span class="lab"><font color="Green" title="E47">A75</font></span></a>, <a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="txt" href="#E5:59_6"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="ref" href="funct_5.html#T39" title="FUNCT_5:th.39">FUNCT_5:39</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">j</font> = <span class="p1">(<span class="default"><a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> as   <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="funct_5.html#T21" title="FUNCT_5:th.21">FUNCT_5:21</a></span>;<br/><a NAME="E4:59_6_1"/><span class="lab"><font color="Green" title="E53">A81</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p3">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D3" title="FUNCT_5:def.3">FUNCT_5:def 3</a></span>;<br/><a NAME="E5:59_6_1"/><span class="lab"><font color="Green" title="E54">A82</font></span>: 
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T4" target="_self" title="FUNCT_6:th.4">Th13</a></span>;<br/><a NAME="E6:59_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A83</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>, the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59_6_1"><span class="lab"><font color="Green" title="E52">A80</font></span></a>, <a class="ref" href="funct_4.html#T42" title="FUNCT_4:th.42">FUNCT_4:42</a></span>;<br/><a NAME="E7:59_6_1"/><span class="lab"><font color="Green" title="E56">A84</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">v</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:59_6_1_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">v</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">v</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">j</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">b</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="c11">j</font> or <font color="Maroon" title="c12">b</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="c9">v</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_6_1_1_1"/>
<font color="Maroon" title="c12">b</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="c11">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b</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="c9">v</font></span><br/>

<a NAME="E2:59_6_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="txt" href="#E5:59_6_1"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E6:59_6_1"><span class="lab"><font color="Green" title="E55">A83</font></span></a>, <a class="txt" href="funct_6.html#E60"><span class="lab"><font color="Green" title="E33">Lm4</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c14">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c15">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:59_6_1_1_1"/><span class="lab"><font color="Green" title="E57">A85</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c15">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:59_6_1_1_1"/>
<font color="Maroon" title="c13">a1</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="c5">s</font>
 <span class="kw">and </span><br/><a NAME="E6:59_6_1_1_1"/><span class="lab"><font color="Green" title="E58">A86</font></span>: 
( <font color="Maroon" title="c14">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> &amp; <font color="Maroon" title="c15">a2</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="c14">g1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></span>;<br/>
<a NAME="E7:59_6_1_1_1"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6_1_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:59_6_1_1_1"/>
<font color="Maroon" title="c12">b</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="c9">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="txt" href="#E4:59_6_1_1_1"><span class="lab"><font color="Green" title="E57">A85</font></span></a>, <a class="txt" href="#E6:59_6_1_1_1"><span class="lab"><font color="Green" title="E58">A86</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c12">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">b</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="c9">v</font> or <font color="Maroon" title="c12">b</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="c11">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:59_6_1_1"/>
<font color="Maroon" title="c12">b</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="c9">v</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">b</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="c11">j</font></span><br/>

<a NAME="E3:59_6_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E1:59_6"><span class="lab"><font color="Green" title="E47">A75</font></span></a>, <a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="ref" href="funct_5.html#T38" title="FUNCT_5:th.38">FUNCT_5:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_6_1_1"/>
<font color="Maroon" title="c12">b</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="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="txt" href="#E5:59_6_1"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="ref" href="funct_5.html#T20" title="FUNCT_5:th.20">FUNCT_5:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_6_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">b</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> implies <font color="Maroon" title="c11">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:59_6_1_2"/>
<font color="Maroon" title="c12">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font></span><br/><a NAME="E2:59_6_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A87</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_6"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="txt" href="#E4:59_6_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="txt" href="#E5:59_6_1"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E6:59_6_1"><span class="lab"><font color="Green" title="E55">A83</font></span></a>, <a class="txt" href="#E7:59_6_1"><span class="lab"><font color="Green" title="E56">A84</font></span></a>, <a class="txt" href="funct_6.html#E60"><span class="lab"><font color="Green" title="E33">Lm4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">a1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c14">g1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c15">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:59_6_1_2"/><span class="lab"><font color="Green" title="E58">A88</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">a1</font>,<font color="Maroon" title="c15">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:59_6_1_2"/><span class="lab"><font color="Green" title="E59">A89</font></span>: 
( <font color="Maroon" title="c13">a1</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="c5">s</font> &amp; <font color="Maroon" title="c14">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">a1</font> &amp; <font color="Maroon" title="c15">a2</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="c14">g1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D2" title="FUNCT_5:def.2">FUNCT_5:def 2</a></span>;<br/><a NAME="E6:59_6_1_2"/><span class="lab"><font color="Green" title="E60">A90</font></span>: 
<font color="Maroon" title="c11">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">z</font>,<font color="Maroon" title="c12">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6_1"><span class="lab"><font color="Green" title="E53">A81</font></span></a>, <a class="txt" href="#E5:59_6_1"><span class="lab"><font color="Green" title="E54">A82</font></span></a>, <a class="txt" href="#E2:59_6_1_2"><span class="lab"><font color="Green" title="E57">A87</font></span></a>, <a class="ref" href="funct_5.html#T20" title="FUNCT_5:th.20">FUNCT_5:20</a></span>;<br/><a NAME="E7:59_6_1_2"/>
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp; <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6_1_2"><span class="lab"><font color="Green" title="E58">A88</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/><span class="kw">hence </span><a NAME="E8:59_6_1_2"/>
<font color="Maroon" title="c11">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="txt" href="#E5:59_6_1_2"><span class="lab"><font color="Green" title="E59">A89</font></span></a>, <a class="txt" href="#E6:59_6_1_2"><span class="lab"><font color="Green" title="E60">A90</font></span></a>, <a class="ref" href="funct_5.html#T38" title="FUNCT_5:th.38">FUNCT_5:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:59_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_6"><span class="lab"><font color="Green" title="E49">A77</font></span></a>, <a class="txt" href="#E7:59_6_1"><span class="lab"><font color="Green" title="E56">A84</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/><span class="kw">hence </span><a NAME="E10:59_6_1"/>
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:59"><span class="lab"><font color="Green" title="E43">A71</font></span></a>, <a class="txt" href="#E21:59"><span class="lab"><font color="Green" title="E46">A74</font></span></a>, <a class="txt" href="#E1:59_6"><span class="lab"><font color="Green" title="E47">A75</font></span></a>, <a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="txt" href="#E1:59_6_1"><span class="lab"><font color="Green" title="E51">A79</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E8:59_6"/>
( <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies ( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:59"><span class="lab"><font color="Green" title="E46">A74</font></span></a>, <a class="txt" href="#E1:59_6"><span class="lab"><font color="Green" title="E47">A75</font></span></a>, <a class="txt" href="#E4:59_6"><span class="lab"><font color="Green" title="E48">A76</font></span></a>, <a class="txt" href="#E5:59_6"><span class="lab"><font color="Green" title="E49">A77</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:59_6"/>
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:59_6"><span class="lab"><font color="Green" title="E50">A78</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E23:59"/><span class="kw">then </span>
<font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:59"><span class="lab"><font color="Green" title="E39">A46</font></span></a>, <a class="txt" href="#E20:59"><span class="lab"><font color="Green" title="E45">A73</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:59"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59"><span class="lab"><font color="Green" title="E36">A43</font></span></a>, <a class="txt" href="#E16:59"><span class="lab"><font color="Green" title="E42">A70</font></span></a>, <a class="txt" href="#E19:59"><span class="lab"><font color="Green" title="E44">A72</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
