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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">Z</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1"/>
( <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_1"><b>suppose </b></a><a NAME="E1:7_1_1"/>
<font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">F</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">D</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>take </b>
<font color="Maroon">F</font>
;<br/><b>thus </b><a NAME="E3:7_1_1"/>
<font color="Maroon">F</font> is <a href="waybel27.html#V2" target="_self">currying</a>
 ;<br/><b>assume </b><a NAME="E4:7_1_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>thus </b><a NAME="E5:7_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2"><b>suppose </b></a><a NAME="E1:7_1_2"/><i><font color="Green">E38</font></i>: 
not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:7_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="fraenkel.html#D1">FRAENKEL:def 1</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="funct_5.html#K3">curry</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">F</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">E</font><b> such that </b><br/><a NAME="E5:7_1_2"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> holds  <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">d</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5">PBOOLE:sch 5</a>();<br/></i><b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">D</font> ;<br/><b>take </b>
<font color="Maroon">F1</font>
;<br/><b>thus </b><a NAME="E7:7_1_2"/>
<font color="Maroon">F1</font> is <a href="waybel27.html#V2" target="_self">currying</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel27.html#D2" target="_self">WAYBEL27:def 2</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font>
 ;<br/><a NAME="E2:7_1_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:7_1_2_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">x1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/><b>thus </b><a NAME="E5:7_1_2_1_1"/>
<font color="Maroon">x</font> is   <a href="funct_1.html#NM1">Function</a>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">R</font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> as   <a href="relat_1.html#NM1">Relation</a> <b>by </b><i><a class="ref" href="relat_1.html#T6">RELAT_1:6</a></i>;<br/><a NAME="E7:7_1_2_1_1"/>
 <a href="funct_5.html#K1">proj1</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="funct_5.html#T21">FUNCT_5:21</a></i>;<br/><b>hence </b><a NAME="E8:7_1_2_1_1"/>
 <a href="funct_5.html#K1">proj1</a> <font color="Maroon">x</font> is   <a href="relat_1.html#NM1">Relation</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T3">RELAT_1:3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">f</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>

<b>assume </b><a NAME="E2:7_1_2_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">f</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>thus </b><font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">d</font>

<br/>.= 
 <a href="funct_5.html#K3">curry</a> <font color="Maroon">f</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E8:7_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>thus </b><a NAME="E9:7_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:7_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">E</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:7_1_2_2"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3">curry</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">Z</font> )
 
<b>by </b><i>, , , <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:7_1_2_2_1"/>
( <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:7_1_2_2_1_1"/><b>then </b>
<font color="Maroon">d</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/><a NAME="E3:7_1_2_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T57">PARTFUN1:57</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_2_2_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:7_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon">y</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>, <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font>
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T55">FUNCT_2:55</a>, <a class="ref" href="funct_5.html#T49">FUNCT_5:49</a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/><b>hence </b><a NAME="E3:7_1_2_2_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:7_1_2_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T113">ZFMISC_1:113</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="7_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:7_1_2_2_1_2"/>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:7_1_2_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_6.html#T19">FUNCT_6:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div>
