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<b>let </b><font color="Maroon">D1</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">D2</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:12_1_2"/><i><font color="Green">E50</font></i>: 
( ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Olive">x</font> <a href="zf_lang.html#K4" target="_self">'='</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> ) ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> holds <br/> <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> holds <br/><font color="Olive">p</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a><br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> holds <br/> <a href="zf_lang.html#K8" target="_self">All</a> <font color="Olive">x</font>,<font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D1</font> ) &amp; ( for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Olive">x</font> <a href="zf_lang.html#K4" target="_self">'='</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/> <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/><font color="Olive">p</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a><br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/> <a href="zf_lang.html#K8" target="_self">All</a> <font color="Olive">x</font>,<font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) holds <br/><font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D</font> ) )
 <b>and </b><br/><a NAME="E2:12_1_2"/><i><font color="Green">E51</font></i>: 
( ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Olive">x</font> <a href="zf_lang.html#K4" target="_self">'='</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> holds <br/> <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> holds <br/><font color="Olive">p</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a><br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> holds <br/> <a href="zf_lang.html#K8" target="_self">All</a> <font color="Olive">x</font>,<font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D2</font> ) &amp; ( for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>   st ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/><font color="Olive">a</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  ) &amp; ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <font color="Olive">x</font> <a href="zf_lang.html#K4" target="_self">'='</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">x</font> <a href="zf_lang.html#K5" target="_self">'in'</a> <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) ) &amp; ( for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/> <a href="zf_lang.html#K6" target="_self">'not'</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) &amp; ( for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/><font color="Olive">p</font> <a href="zf_lang.html#K7" target="_self">'&amp;'</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) &amp; ( for <font color="Olive">x</font> being  <a href="zf_lang.html#NM1">Variable</a><br/> for <font color="Olive">p</font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/> <a href="zf_lang.html#K8" target="_self">All</a> <font color="Olive">x</font>,<font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> ) holds <br/><font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">D</font> ) )
 ;<br/>

<a NAME="E3:12_1_2"/>
( <font color="Maroon">D1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D2</font> &amp; <font color="Maroon">D2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:12_1_2"/>
<font color="Maroon">D1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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