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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="funct_1.html#NM1">Function</a>;<br/><b>let </b><font color="Maroon">f2</font> be   <a href="funct_1.html#NM1">Function</a>;<br/>







<b>assume </b><a NAME="E1:45"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</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#K1">dom</a> <font color="Maroon">f2</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="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f2</font> )
 ;<br/>

<a NAME="E2:45"/><b>then </b><i><font color="Green">E32</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>  implies ( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 
<b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E34</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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>
 ;<br/><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/><b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:45_1_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> )
 <b>and </b><br/><a NAME="E5:45_1_1"/><i><font color="Green">E38</font></i>: 
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> <font color="Maroon">Y</font> holds <br/><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Olive">y</font>
 <b>by </b><i>, , </i>;<br/><b>consider </b><font color="Maroon">g2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><a href="funct_5.html#K3" target="_self">curry</a> <font color="Maroon">f2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">g2</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f2</font> )
 <b>and </b><br/><a NAME="E8:45_1_1"/><i><font color="Green">E40</font></i>: 
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> <font color="Maroon">Y</font> holds <br/><font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<font color="Olive">y</font>
 <b>by </b><i>, , <a class="txt" href="funct_5.html#E2:45_1_1"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E9:45_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">f1</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i><a class="txt" href="funct_5.html#E2:45_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="funct_5.html#E5:45_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E10:45_1_1"/>
<font color="Maroon">f2</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a> </span>)</span>
 <b>by </b><i>, <a class="txt" href="funct_5.html#E2:45_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="funct_5.html#E4:45_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="funct_5.html#E7:45_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="funct_5.html#E8:45_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>hence </b><a NAME="E11:45_1_1"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="funct_5.html#D2" target="_self">Def2</a>, <a class="txt" href="funct_5.html#E2:45_1_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:45_1"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/>


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