<?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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">X</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>  holds <br/> <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">f</font></span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Olive" title="b1">f</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</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>  &amp; <font color="Maroon" title="c1">X</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  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</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> 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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b1">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> );<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:89"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c2">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E2:89"/><span class="lab"><font color="Green" title="E50">A2</font></span>: 
<font color="Maroon" title="c1">X</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="Red">::  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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>, <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> </span><br/>

<a NAME="E3:89"/><span class="lab"><font color="Green" title="E51">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">y</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">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> implies  ex <font color="Olive" title="b1">y</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">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_1"/>
<font color="Maroon" title="c3">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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<a NAME="E2:89_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">g</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> &amp; ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> holds <br/><font color="Olive" title="b1">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</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/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">g</font> = <font color="Maroon" title="c3">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y</font>]</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c4">g</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E5:89_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">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="E5:89"/><span class="lab"><font color="Green" title="E52">A4</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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> &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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> 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="#E3:89"><span class="lab"><font color="Green" title="E51">A3</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c3">F</font>
; <span class="comment"><font color="Red">:: 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="Red">::  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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> &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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<a NAME="E6:89"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">g</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> holds <br/>(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b1">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">g</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b1">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">g</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> implies (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font> ) )</span><br/>

<a NAME="E1:89_2"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c4">g</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> <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="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T169" title="RELAT_1:th.169">RELAT_1:169</a></span>;<br/>
<a NAME="E2:89_2"/><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <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="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:89_2"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
<font color="Maroon" title="c4">g</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font> )</span><br/>

<a NAME="E4:89_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A9</font></span>: 
 <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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</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="c4">g</font> implies <font color="Maroon" title="c5">a</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><span class="kw">assume </span><a NAME="E1:89_2_1"/>
<font color="Maroon" title="c5">a</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="c4">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:89_2_1"/><span class="lab"><font color="Green" title="E59">A11</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> <font color="Maroon" title="c4">g</font>
 <span class="kw">and </span><br/><a NAME="E4:89_2_1"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E5:89_2_1"/>
( <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_2"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E3:89_2_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">hence </span><a NAME="E6:89_2_1"/>
<font color="Maroon" title="c5">a</font> is   <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="#E4:89_2_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:89_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A13</font></span>: 
 <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="funct_6.html#E30"><span class="lab"><font color="Green" title="E18">Lm1</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E60">A14</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">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 <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="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <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="c5">z</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"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:89_2_2"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> <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="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <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="c5">z</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"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span></span><br/><a NAME="E2:89_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> &amp; <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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> <font color="Maroon" title="c1">X</font>,<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="c5">z</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_2"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><a NAME="E3:89_2_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  <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> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">h</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="c5">z</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="E4:89_2_2"/>
<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="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <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="c5">z</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"><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_2_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:89_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">X</font></span>)</span> <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="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E9:89_2"/><span class="kw">then </span>
 <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E6:89_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E1:89_2"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E7:89_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="funct_6.html#D2" target="_self" title="FUNCT_6:def.2">Def2</a></span>;<br/>
<a NAME="E10:89_2"/><span class="kw">then </span>
 <a href="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="funct_6.html#T43" target="_self" title="FUNCT_6:th.43">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:89_2"/><span class="lab"><font color="Green" title="E61">A16</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">g</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89_2"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="txt" href="funct_6.html#E30"><span class="lab"><font color="Green" title="E18">Lm1</font></span></a>, <a class="ref" href="funct_6.html#T49" target="_self" title="FUNCT_6:th.49">Th49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> or <font color="Maroon" title="c5">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> <font color="Maroon" title="c2">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E12:89_2"/>
<font color="Maroon" title="c5">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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">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> <font color="Maroon" title="c2">f</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:89_2"/><span class="lab"><font color="Green" title="E62">A17</font></span>: 
<font color="Maroon" title="c6">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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E15:89_2"/><span class="lab"><font color="Green" title="E63">A18</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c4">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c5">y</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="#E14:89_2"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E15:89_2"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="funct_6.html#T50" target="_self" title="FUNCT_6:th.50">Th50</a></span>;<br/>
<a NAME="E17:89_2"/><span class="lab"><font color="Green" title="E64">A19</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">by</span> <span class="lab"><a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E6:89_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E1:89_2"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="txt" href="#E14:89_2"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E15:89_2"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_3"><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="Red">::  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="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> <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="E1:89_2_3"/><span class="lab"><font color="Green" title="E65">A20</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="Red">::  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> <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/><a NAME="E2:89_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A21</font></span>: 
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">g</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="c6">x</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="#E14:89_2"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E15:89_2"><span class="lab"><font color="Green" title="E63">A18</font></span></a>, <a class="txt" href="#E17:89_2"><span class="lab"><font color="Green" title="E64">A19</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>;<br/><a NAME="E3:89_2_3"/>
( <font color="Maroon" title="c4">g</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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> <font color="Maroon" title="c1">X</font>,<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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_2"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E1:89_2_3"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">j</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:89_2_3"/><span class="lab"><font color="Green" title="E67">A22</font></span>: 
<font color="Maroon" title="c4">g</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">j</font>
 <span class="kw">and </span><br/><a NAME="E6:89_2_3"/><span class="lab"><font color="Green" title="E68">A23</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E7:89_2_3"/><span class="lab"><font color="Green" title="E69">A24</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">j</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="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/><a NAME="E8:89_2_3"/><span class="lab"><font color="Green" title="E70">A25</font></span>: 
<font color="Maroon" title="c9">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</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="c9">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89_2"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E14:89_2"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E6:89_2_3"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:89_2_3"/>
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">g</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="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_2"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="txt" href="#E2:89_2"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E11:89_2"><span class="lab"><font color="Green" title="E61">A16</font></span></a>, <a class="txt" href="#E14:89_2"><span class="lab"><font color="Green" title="E62">A17</font></span></a>, <a class="txt" href="#E1:89_2_3"><span class="lab"><font color="Green" title="E65">A20</font></span></a>, <a class="txt" href="#E5:89_2_3"><span class="lab"><font color="Green" title="E67">A22</font></span></a>, <a class="txt" href="#E6:89_2_3"><span class="lab"><font color="Green" title="E68">A23</font></span></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></span>;<br/><span class="kw">hence </span><a NAME="E10:89_2_3"/>
<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> <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="#E7:89_2_3"><span class="lab"><font color="Green" title="E69">A24</font></span></a>, <a class="txt" href="#E2:89_2_3"><span class="lab"><font color="Green" title="E66">A21</font></span></a>, <a class="txt" href="#E8:89_2_3"><span class="lab"><font color="Green" title="E70">A25</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E19:89_2"/>
<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:89_2"><span class="lab"><font color="Green" title="E64">A19</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E7:89"/>
<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="Red">::  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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> &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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><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="Red">:: according to </font></span><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">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="Red">::  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="E1:89_3"/><span class="lab"><font color="Green" title="E54">A26</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="E2:89_3"/><span class="lab"><font color="Green" title="E55">A27</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="E3:89_3"/><span class="lab"><font color="Green" title="E56">A28</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>
 <span class="kw">and </span><br/><a NAME="E4:89_3"/><span class="lab"><font color="Green" title="E57">A29</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">gy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:89_3"/><span class="lab"><font color="Green" title="E58">A30</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">gy</font>
 <span class="kw">and </span><br/><a NAME="E7:89_3"/><span class="lab"><font color="Green" title="E59">A31</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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c6">gy</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:89_3"><span class="lab"><font color="Green" title="E55">A27</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">gx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:89_3"/><span class="lab"><font color="Green" title="E60">A32</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">gx</font>
 <span class="kw">and </span><br/><a NAME="E10:89_3"/><span class="lab"><font color="Green" title="E61">A33</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> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">gx</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E1:89_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a></span>;<br/>
<a NAME="E11:89_3"/><span class="lab"><font color="Green" title="E62">A34</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">gx</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E1:89_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E9:89_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<a NAME="E12:89_3"/><span class="lab"><font color="Green" title="E63">A35</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <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="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a></span>;<br/>
<a NAME="E13:89_3"/><span class="lab"><font color="Green" title="E64">A36</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">gy</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:89_3"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E6:89_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_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="Red">::  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="c7">gx</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="c6">gy</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="E1:89_3_1"/><span class="lab"><font color="Green" title="E65">A37</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">gx</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="c6">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/><a NAME="E2:89_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A38</font></span>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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> <font color="Maroon" title="c1">X</font>,<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a></span>;<br/><a NAME="E3:89_3_1"/>
<font color="Maroon" title="c6">gy</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E2:89_3"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="txt" href="#E6:89_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="txt" href="#E1:89_3_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">hy</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:89_3_1"/><span class="lab"><font color="Green" title="E67">A39</font></span>: 
( <font color="Maroon" title="c6">gy</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">hy</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">hy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">and </span><br/><a NAME="E6:89_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">hy</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_3_1"><span class="lab"><font color="Green" title="E66">A38</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="E7:89_3_1"/>
<font color="Maroon" title="c7">gx</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E1:89_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E9:89_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="txt" href="#E1:89_3_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">hx</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E9:89_3_1"/><span class="lab"><font color="Green" title="E68">A40</font></span>: 
( <font color="Maroon" title="c7">gx</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">hx</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">hx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> )
 <span class="kw">and </span><br/><a NAME="E10:89_3_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">hx</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="kw">by</span> <span class="lab"><a class="txt" href="#E2:89_3_1"><span class="lab"><font color="Green" title="E66">A38</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="E11:89_3_1"/><span class="lab"><font color="Green" title="E69">A41</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">gx</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E1:89_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E9:89_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a></span>;<br/><a NAME="E12:89_3_1"/><span class="lab"><font color="Green" title="E70">A42</font></span>: 
(  <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">gx</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">gx</font> &amp; <font color="Maroon" title="c7">gx</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">gx</font></span>)</span> <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="c7">gx</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E1:89_3"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E9:89_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="ref" href="relat_1.html#T169" title="RELAT_1:th.169">RELAT_1:169</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3_1_1"><span class="kw">now </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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c10">hx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">hy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> )</span><br/><span class="kw">assume </span><a NAME="E1:89_3_1_1"/><span class="lab"><font color="Green" title="E71">A43</font></span>: 
<font color="Maroon" title="c11">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">hx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">hy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">fx</font> = <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c7">gx</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</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="#E11:89_3_1"><span class="lab"><font color="Green" title="E69">A41</font></span></a>, <a class="ref" href="funct_6.html#T50" target="_self" title="FUNCT_6:th.50">Th50</a></span>;<br/><a NAME="E3:89_3_1_1"/><span class="lab"><font color="Green" title="E72">A44</font></span>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c7">gx</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="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">hx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> &amp; <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c6">gy</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="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">hy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89_3"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="txt" href="#E13:89_3"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="txt" href="#E1:89_3_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E9:89_3_1"><span class="lab"><font color="Green" title="E68">A40</font></span></a>, <a class="txt" href="#E5:89_3_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="txt" href="#E1:89_3_1_1"><span class="lab"><font color="Green" title="E71">A43</font></span></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></span>;<br/><a NAME="E4:89_3_1_1"/><span class="lab"><font color="Green" title="E73">A45</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">fx</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="c7">gx</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89_3_1"><span class="lab"><font color="Green" title="E69">A41</font></span></a>, <a class="txt" href="#E12:89_3_1"><span class="lab"><font color="Green" title="E70">A42</font></span></a>, <a class="txt" href="#E1:89_3_1_1"><span class="lab"><font color="Green" title="E71">A43</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>;<br/><a NAME="E5:89_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">fx</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> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c7">gx</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="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:89_3"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="txt" href="#E1:89_3_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E11:89_3_1"><span class="lab"><font color="Green" title="E69">A41</font></span></a>, <a class="txt" href="#E1:89_3_1_1"><span class="lab"><font color="Green" title="E71">A43</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>;<br/><span class="kw">hence </span><a NAME="E6:89_3_1_1"/>
<font color="Maroon" title="c10">hx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">hy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_3"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="txt" href="#E10:89_3"><span class="lab"><font color="Green" title="E61">A33</font></span></a>, <a class="txt" href="#E7:89_3"><span class="lab"><font color="Green" title="E59">A31</font></span></a>, <a class="txt" href="#E11:89_3"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="txt" href="#E1:89_3_1"><span class="lab"><font color="Green" title="E65">A37</font></span></a>, <a class="txt" href="#E11:89_3_1"><span class="lab"><font color="Green" title="E69">A41</font></span></a>, <a class="txt" href="#E1:89_3_1_1"><span class="lab"><font color="Green" title="E71">A43</font></span></a>, <a class="txt" href="#E4:89_3_1_1"><span class="lab"><font color="Green" title="E73">A45</font></span></a>, <a class="txt" href="#E3:89_3_1_1"><span class="lab"><font color="Green" title="E72">A44</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E14:89_3_1"/>
<font color="Maroon" title="c7">gx</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="c6">gy</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="#E9:89_3_1"><span class="lab"><font color="Green" title="E68">A40</font></span></a>, <a class="txt" href="#E5:89_3_1"><span class="lab"><font color="Green" title="E67">A39</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:89_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:89_3"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E9:89_3"><span class="lab"><font color="Green" title="E60">A32</font></span></a>, <a class="txt" href="#E6:89_3"><span class="lab"><font color="Green" title="E58">A30</font></span></a>, <a class="txt" href="#E11:89_3"><span class="lab"><font color="Green" title="E62">A34</font></span></a>, <a class="txt" href="#E13:89_3"><span class="lab"><font color="Green" title="E64">A36</font></span></a>, <a class="txt" href="#E12:89_3"><span class="lab"><font color="Green" title="E63">A35</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">thus </span><a NAME="E8:89"/>
 <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="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E9:89"/>
 <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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>
  <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></a><span class="hide">  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</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#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="89_4"><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="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_4"/>
<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="Red">::  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="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</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="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:89_4"/><span class="lab"><font color="Green" title="E54">A46</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="E4:89_4"/><span class="lab"><font color="Green" title="E55">A47</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#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">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="E6:89_4"/><span class="lab"><font color="Green" title="E56">A48</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="E7:89_4"/><span class="lab"><font color="Green" title="E57">A49</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E3:89_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E4:89_4"><span class="lab"><font color="Green" title="E55">A47</font></span></a></span>;<br/>
<a NAME="E8:89_4"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c6">g</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E3:89_4"><span class="lab"><font color="Green" title="E54">A46</font></span></a>, <a class="txt" href="#E6:89_4"><span class="lab"><font color="Green" title="E56">A48</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:89_4"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:89_4"><span class="lab"><font color="Green" title="E57">A49</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">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="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span> or <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">proj2</a> <font color="Maroon" title="c3">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:89"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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">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">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="E12:89"/><span class="lab"><font color="Green" title="E54">A50</font></span>: 
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font>
 <span class="kw">and </span><br/><a NAME="E13:89"/><span class="lab"><font color="Green" title="E55">A51</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E14:89"/><span class="lab"><font color="Green" title="E56">A52</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</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> <font color="Maroon" title="c2">f</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">set </span><font color="Maroon" title="c6">g</font> = <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>;<br/>
<a NAME="E15:89"/><span class="lab"><font color="Green" title="E57">A53</font></span>: 
<font color="Maroon" title="c5">h</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font></span>)</span> <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="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T169" title="RELAT_1:th.169">RELAT_1:169</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E58">A54</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_5"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:89_5"/><span class="lab"><font color="Green" title="E59">A55</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font></span><br/><a NAME="E2:89_5"/><span class="kw">then </span>
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</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="c5">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E3:89_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A56</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="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</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>  <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="c2">f</font> holds <br/><font color="Olive" title="b1">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</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="Olive" title="b2">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> = 
 <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="#E1:89_5"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</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="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E1:89_5"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="txt" href="#E3:89_5"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="ref" href="funct_6.html#T31" target="_self" title="FUNCT_6:th.31">Th31</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E17:89"/><span class="lab"><font color="Green" title="E59">A57</font></span>: 
 <a href="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font></span>)</span> <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"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T49" target="_self" title="FUNCT_6:th.49">Th49</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_6"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">z</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="c5">h</font> implies <font color="Maroon" title="c7">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><span class="kw">assume </span><a NAME="E1:89_6"/>
<font color="Maroon" title="c7">z</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="c5">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><a NAME="E2:89_6"/><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="c7">z</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>  <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">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="c2">f</font> holds <br/><font color="Olive" title="b1">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</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="Olive" title="b2">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E3:89_6"/>
<font color="Maroon" title="c7">z</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E19:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A58</font></span>: 
 <a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="funct_6.html#E30"><span class="lab"><font color="Green" title="E18">Lm1</font></span></a></span>;<br/>
<a NAME="E20:89"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> <a href="relat_1.html#K10" title="RELAT_1:func.10">"</a> <span class="p1">(<span class="default"><a href="funct_6.html#K1" title="FUNCT_6:func.1">SubFuncs</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <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="c5">h</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D2" target="_self" title="FUNCT_6:def.2">Def2</a>, <a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E21:89"/><span class="kw">then </span>
<font color="Maroon" title="c1">X</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E19:89"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E15:89"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E16:89"><span class="lab"><font color="Green" title="E58">A54</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<a NAME="E22:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A59</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <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="#E2:89"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="txt" href="#E17:89"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="ref" href="funct_6.html#T43" target="_self" title="FUNCT_6:th.43">Th43</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E62">A60</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="89_7"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> <font color="Maroon" title="c2">f</font> implies <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:89_7"/><span class="lab"><font color="Green" title="E63">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> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">i</font> = <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="ref" href="funct_6.html#T50" target="_self" title="FUNCT_6:th.50">Th50</a></span>;<br/><a NAME="E3:89_7"/><span class="lab"><font color="Green" title="E64">A62</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E19:89"><span class="lab"><font color="Green" title="E60">A58</font></span></a>, <a class="txt" href="#E15:89"><span class="lab"><font color="Green" title="E57">A53</font></span></a>, <a class="txt" href="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E1:89_7"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>;<br/><a NAME="E4:89_7"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">i</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="c7">z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</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="c8">i</font> or <font color="Maroon" title="c9">x</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="c7">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:89_7_1"/>
<font color="Maroon" title="c9">x</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="c8">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</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="c7">z</font></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> <span class="kw"> such that </span><br/><a NAME="E3:89_7_1"/><span class="lab"><font color="Green" title="E65">A63</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="c8">i</font>
 <span class="kw">and </span><br/><a NAME="E4:89_7_1"/><span class="lab"><font color="Green" title="E66">A64</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E5:89_7_1"/>
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</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="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E3:89_7"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="txt" href="#E3:89_7_1"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">j</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E7:89_7_1"/><span class="lab"><font color="Green" title="E67">A65</font></span>: 
( <font color="Maroon" title="c5">h</font> <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> <font color="Maroon" title="c11">j</font> &amp;  <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="c2">f</font> )
 <span class="kw">and </span><br/><a NAME="E8:89_7_1"/><span class="lab"><font color="Green" title="E68">A66</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><font color="Maroon" title="c11">j</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<font color="Maroon" title="c8">i</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> = 
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">a</font>,<font color="Maroon" title="c7">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E1:89_7"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="txt" href="#E3:89_7_1"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="ref" href="funct_6.html#T51" target="_self" title="FUNCT_6:th.51">Th51</a></span>
<br/>.= 
<font color="Maroon" title="c11">j</font> <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="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E1:89_7"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="txt" href="#E3:89_7"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="txt" href="#E3:89_7_1"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="txt" href="#E7:89_7_1"><span class="lab"><font color="Green" title="E67">A65</font></span></a>, <a class="ref" href="funct_5.html#T45" title="FUNCT_5:th.45">FUNCT_5:45</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E10:89_7_1"/>
<font color="Maroon" title="c9">x</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="c7">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89_7"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="txt" href="#E4:89_7_1"><span class="lab"><font color="Green" title="E66">A64</font></span></a>, <a class="txt" href="#E8:89_7_1"><span class="lab"><font color="Green" title="E68">A66</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:89_7"/><span class="kw">then </span>
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:89_7"><span class="lab"><font color="Green" title="E64">A62</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">hence </span><a NAME="E6:89_7"/>
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</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="txt" href="#E1:89_7"><span class="lab"><font color="Green" title="E63">A61</font></span></a>, <a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E24:89"/><span class="lab"><font color="Green" title="E63">A67</font></span>: 
 <a href="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span> <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"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T49" target="_self" title="FUNCT_6:th.49">Th49</a></span>;<br/>
<a NAME="E25:89"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#T10" target="_self" title="FUNCT_6:th.10">Th10</a></span>;<br/>
<a NAME="E26:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A68</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:89"><span class="lab"><font color="Green" title="E56">A52</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E27:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A69</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#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</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="ref" href="funct_5.html#T33" title="FUNCT_5:th.33">FUNCT_5:33</a></span>;<br/>
<a NAME="E28:89"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_6.html#D9" target="_self" title="FUNCT_6:def.9">Def9</a></span>;<br/>
<a NAME="E29:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A70</font></span>: 
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <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#K9" title="FUNCT_6:func.9">Funcs</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E23:89"><span class="lab"><font color="Green" title="E62">A60</font></span></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></span>;<br/>
<a NAME="E30:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A71</font></span>: 
 ex <font color="Olive" title="b1">g9</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g9</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Olive" title="b1">g9</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a></span>;<br/>
<a NAME="E31:89"/>
 <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">h</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font></span>)</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="#E26:89"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="ref" href="funct_5.html#T33" title="FUNCT_5:th.33">FUNCT_5:33</a></span>;<br/>
<a NAME="E32:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A72</font></span>: 
<span class="p1">(<span class="default"><a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:89"><span class="lab"><font color="Green" title="E59">A57</font></span></a>, <a class="txt" href="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>;<br/>
<a NAME="E33:89"/><span class="lab"><font color="Green" title="E69">A73</font></span>: 
(  <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <font color="Maroon" title="c5">h</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">h</font></span>)</span> &amp;  <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">h</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K3" title="FUNCT_5:func.3">curry'</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">h</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#D5" title="FUNCT_5:def.5">FUNCT_5:def 5</a>, <a class="ref" href="funct_5.html#D6" title="FUNCT_5:def.6">FUNCT_5:def 6</a></span>;<br/>
<a NAME="E34:89"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:89"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E29:89"><span class="lab"><font color="Green" title="E66">A70</font></span></a></span>;<br/>
<a NAME="E35:89"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A74</font></span>: 
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">h</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</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="funct_6.html#K4" title="FUNCT_6:func.4">meet</a> <span class="p3">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p4"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p3"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></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">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:89"><span class="lab"><font color="Green" title="E55">A51</font></span></a>, <a class="txt" href="#E22:89"><span class="lab"><font color="Green" title="E61">A59</font></span></a>, <a class="txt" href="#E27:89"><span class="lab"><font color="Green" title="E65">A69</font></span></a>, <a class="txt" href="#E24:89"><span class="lab"><font color="Green" title="E63">A67</font></span></a>, <a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></span>;<br/>
<a NAME="E36:89"/>
 <a href="funct_5.html#K4" title="FUNCT_5:func.4">uncurry'</a> <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#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c5">h</font></span>)</span></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">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:89"><span class="lab"><font color="Green" title="E65">A69</font></span></a>, <a class="ref" href="funct_5.html#T56" title="FUNCT_5:th.56">FUNCT_5:56</a></span>;<br/>
<a NAME="E37:89"/><span class="kw">then </span>
<span class="p1"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><span class="p2"><a href="funct_6.html#K6" title="FUNCT_6:func.6">&lt;:</a><span class="default"><font color="Maroon" title="c5">h</font></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span></span><a href="funct_6.html#K6" title="FUNCT_6:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:89"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="txt" href="#E26:89"><span class="lab"><font color="Green" title="E64">A68</font></span></a>, <a class="txt" href="#E32:89"><span class="lab"><font color="Green" title="E68">A72</font></span></a>, <a class="txt" href="#E35:89"><span class="lab"><font color="Green" title="E70">A74</font></span></a>, <a class="txt" href="#E33:89"><span class="lab"><font color="Green" title="E69">A73</font></span></a>, <a class="ref" href="funct_5.html#T55" title="FUNCT_5:th.55">FUNCT_5:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E38:89"/>
<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">proj2</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:89"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="txt" href="#E12:89"><span class="lab"><font color="Green" title="E54">A50</font></span></a>, <a class="txt" href="#E29:89"><span class="lab"><font color="Green" title="E66">A70</font></span></a>, <a class="txt" href="#E30:89"><span class="lab"><font color="Green" title="E67">A71</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
