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<b>let </b><font color="Maroon">x</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K7">*</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">y</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K7">*</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> holds <br/><font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</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">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Olive">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Olive">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E79</font></i>: 
for <font color="Olive">h</font> being  <a href="funct_1.html#NM1">Function</a>  st <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> holds <br/><font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Olive">h</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">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Olive">h</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Olive">h</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<a NAME="E3:13_1_2"/><i><font color="Green">E80</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E4:13_1_2"/><b>then </b><i><font color="Green">E81</font></i>: 
( <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">r</font> &amp; <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">rj</font> = <font color="Maroon">r</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">pj</font> = <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">I</font> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E7:13_1_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K7">*</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:13_1_2"/><b>then </b><i><font color="Green">E82</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">pj</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K7">*</a> <font color="Maroon">r</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">rj</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E9:13_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">h</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1_2_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="pboole.html#K6">#</a> </span>)</span> <a href="pboole.html#K7">*</a> <font color="Maroon">p</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/><a NAME="E2:13_1_2_1"/><b>then </b>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">pj</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pboole.html#D19">PBOOLE:def 19</a></i>;<br/><b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:13_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:13_1_2_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">pj</font></span>)</span> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</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"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">pj</font></span>)</span> holds <br/><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">B</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">pj</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> ) )
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E6:13_1_2_1"/>
( <font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h1</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">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">h1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">h1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h1</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">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K15">pr1</a> <font color="Maroon">h1</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="mcart_1.html#K16">pr2</a> <font color="Maroon">h1</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E7:13_1_2_1"/>
<font color="Maroon">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:13_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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