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<b>let </b><font color="Maroon" title="c1">f</font> be  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E10">A1</font></i>: 
<font color="Maroon" title="c2">i</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#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font></span>)</span>
 ;<br/>

<i><font color="Green" title="E11">A2</font></i>:  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font> = 
 <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="c1">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_6.html#D12" title="FUNCT_6:def.12">FUNCT_6:def 12</a></i>
<br/>.= 
 <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="c1">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_5.html#D5" title="FUNCT_5:def.5">FUNCT_5:def 5</a></i>
;<br/>
<b>thus </b><a NAME="E3:12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E12">A3</font></i>: 
( <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:12_2"/><i><font color="Green" title="E13">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_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="c1">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="waybel27.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="waybel27.html#E3:12_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_5.html#T38" title="FUNCT_5:th.38">FUNCT_5:38</a></i>;<br/>
<a NAME="E5:12_2"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c1">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_4.html#T43" title="FUNCT_4:th.43">FUNCT_4:43</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c7">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c8">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:12_2"/><i><font color="Green" title="E15">A6</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">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="c1">f</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">g</font> )
 <b>by </b><i><a class="ref" href="funct_5.html#D4" title="FUNCT_5:def.4">FUNCT_5:def 4</a></i>;<br/>
<a NAME="E8:12_2"/>
<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="c1">f</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E7:12_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:12_2"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
<font color="Maroon" title="c1">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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="waybel27.html#E3:12_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E10:12_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<font color="Maroon" title="c4">x</font> = 
<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="c1">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c5">y</font>
<b>by </b><i><a class="txt" href="waybel27.html#E1:12"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="waybel27.html#E2:12"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="waybel27.html#E3:12_2"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="funct_5.html#T38" title="FUNCT_5:th.38">FUNCT_5:38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c2">i</font>
<b>by </b><i><a class="txt" href="waybel27.html#E4:12_2"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="funct_4.html#T44" title="FUNCT_4:th.44">FUNCT_4:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>
<b>by </b><i><a class="txt" href="waybel27.html#E5:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="waybel27.html#E10:12_2"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="funct_5.html#D4" title="FUNCT_5:def.4">FUNCT_5:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E12:12_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c3">A</font></span>)</span>,<font color="Maroon" title="c2">i</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E9:12_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>


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


</div>
