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<div class="add">

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

<b>thus </b><a NAME="E1:37"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:37_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:37_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:37_1"/><i><font color="Green">E32</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2" target="_self">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Olive">y</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:37_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K2" target="_self">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="txt" href="funct_5.html#E3:37_1"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E8:37_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="funct_5.html#E3:37_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


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

<b>let </b><font color="Maroon">t</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="E2:37"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5" target="_self">curry'</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:37"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5" target="_self">curry'</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_5.html#K5" target="_self">curry'</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:37"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="funct_5.html#K5" target="_self">curry'</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3" target="_self">curry</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:37"/><i><font color="Green">E36</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2" target="_self">proj2</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2" target="_self">proj2</a> <span class="p4">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p5">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">z</font>,<font color="Olive">y</font> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E8:37"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K2" target="_self">proj2</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T42">FUNCT_4:42</a></i>;<br/>
<a NAME="E9:37"/><b>then </b>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="funct_5.html#K1" target="_self">proj1</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E10:37"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="funct_5.html#K1" target="_self">proj1</a> <span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#D5">PARTFUN1:def 5</a></i>;<br/>


</div>
