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<b>let </b><font color="Maroon">a1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">a2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:5_1_2"/><i><font color="Green">E30</font></i>: 
 ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; ( for <font color="Olive">H'</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">H'</font>,<font color="Olive">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( <font color="Olive">H'</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">H'</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">H'</font>) ) &amp; ( <font color="Olive">H'</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">H'</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">H'</font>) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>( <a href="zf_lang.html#K23">bound_in</a> <font color="Olive">H'</font>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) ) )
 <b>and </b><br/><a NAME="E2:5_1_2"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">A</font> being   <a href="hidden.html#M1">set</a>  st <br/>( ( for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="zf_lang.html#NM1">Variable</a> holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K4">'='</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">x</font> <a href="zf_lang.html#K5">'in'</a> <font color="Olive">y</font></span>)</span>,<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">x</font>,<font color="Olive">y</font>)</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">H</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; ( for <font color="Olive">H'</font> being  <a href="zf_lang.html#NM2">ZF-formula</a><br/> for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">H'</font>,<font color="Olive">a</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>( ( <font color="Olive">H'</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">H'</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">H'</font>) ) &amp; ( <font color="Olive">H'</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="zf_lang.html#K18">Var1</a> <font color="Olive">H'</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Olive">H'</font>) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V4">negative</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V5">conjunctive</a> implies  ex <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b</font>,<font color="Olive">c</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) &amp; ( <font color="Olive">H'</font> is <a href="zf_lang.html#V6">universal</a> implies  ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>5</sub></font>( <a href="zf_lang.html#K23">bound_in</a> <font color="Olive">H'</font>,<font color="Olive">b</font>) &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H'</font></span>)</span>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> ) ) ) ) )
 ;<br/>

<b>thus </b><a NAME="E3:5_1_2"/>
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>from </b><i><a class="ref" href="zf_model.html#S2" target="_self">ZF_MODEL:sch 2</a>(, );</i><br/>


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