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<b>let </b><font color="Maroon">f</font> be  <a href="funcop_1.html#V1">Function-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">g</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Olive">g</font>
 ;<br/>

<b>let </b><font color="Maroon">x</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="E2:13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K5">pi</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">i</font>
 ;<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="E4:13"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="card_3.html#D6">CARD_3:def 6</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:13"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<i><font color="Green">E48</font></i>:  <a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font> = 
 <a href="funct_5.html#K5">curry'</a> <span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#D12">FUNCT_6:def 12</a></i>
<br/>.= 
 <a href="funct_5.html#K3">curry</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#D5">FUNCT_5:def 5</a></i>
;<br/>
<a NAME="E8:13"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:13"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:13"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <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#K4">uncurry</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>;<br/>
<a NAME="E11:13"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>;<br/>
<a NAME="E12:13"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K1">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> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">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> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_5.html#D1">FUNCT_5:def 1</a>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E13:13"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p5">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E14:13"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</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> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2">(<span class="default"><a href="funct_5.html#K2">proj2</a> <span class="p3">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p4">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p5">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E16:13"/><i><font color="Green">E58</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3">curry</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K2">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> <span class="p4">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></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">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2">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> <span class="p0">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span> &amp; ( for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> holds <br/><font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Olive">b</font> ) )
 <b>by </b><i>, <a class="ref" href="funct_5.html#D3">FUNCT_5:def 3</a></i>;<br/>
<a NAME="E17:13"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="funct_5.html#K2">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> <span class="p4">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></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">i</font></span><a href="tarski.html#K1">}</a></span>,<span class="p3">(<span class="default"><a href="funct_5.html#K2">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> <span class="p0">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E18:13"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="funct_5.html#D2">FUNCT_5:def 2</a></i>;<br/>
<a NAME="E19:13"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">i</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">curry</a> <span class="p2">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p3">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_5.html#D3">FUNCT_5:def 3</a></i>;<br/>
<a NAME="E20:13"/><i><font color="Green">E74</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2">~</a> <span class="p2">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">y</font>
<b>by </b><i>, , , <a class="ref" href="funct_5.html#T38">FUNCT_5:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K4">uncurry</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i>, <a class="ref" href="funct_4.html#D2">FUNCT_4:def 2</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i>, , , , <a class="ref" href="funct_5.html#D4">FUNCT_5:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E22:13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="relat_1.html#K9">.:</a> <font color="Maroon">A</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


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