<?xml version="1.0"?>
<div class="add">

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,  <a href="zf_lang.html#NM1">Variable</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,  <a href="zf_lang.html#NM1">Variable</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(  <a href="zf_lang.html#NM1">Variable</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>6</sub></font>(  <a href="zf_lang.html#NM2">ZF-formula</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<a NAME="E1:4_1"/><i><font color="Green">E35</font></i>: 
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>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>(<font color="Olive">H'</font>) iff  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="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> &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>by </b><i/>;<br/>
<div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>thus </b><a NAME="E1:4_1_1"/>
( ( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H</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="Maroon">H</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font>) ) &amp; ( <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H</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="Maroon">H</font>, <a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font>) ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a> implies <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>6</sub></font>( <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>)) ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a> implies for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>( <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font>) &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>6</sub></font>( <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>) holds <br/><font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) ) &amp; ( <font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a> implies <font color="Maroon">H<sub>6</sub></font>(<font color="Maroon">H</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="Maroon">H</font>,<font color="Maroon">H<sub>6</sub></font>( <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>)) ) )
 <b>from </b><i><a class="ref" href="zf_model.html#S3" target="_self">ZF_MODEL:sch 3</a>();</i><br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2">ZF-formula</a>] <b>means </b><font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">a<sub>1</sub></font>)];<br/>
<a NAME="E3:4_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V7">atomic</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:4_1_2"/>
( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  or <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D15">ZF_LANG:def 15</a><br/>

<a NAME="E2:4_1_2"/><b>then </b>
( <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) or <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K18">Var1</a> <font color="Maroon">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K19">Var2</a> <font color="Maroon">H</font></span>)</span>) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:4_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V4">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:4_1_3"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<a NAME="E2:4_1_3"/><b>then </b>
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font></span>)</span>))
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_3"/>
(  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:4_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V5">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:4_1_4"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<a NAME="E2:4_1_4"/><b>then </b>
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span>),<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font></span>)</span>))
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_4"/>
(  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font>] or  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a>  st <font color="Olive">H</font> is <a href="zf_lang.html#V6">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/>

<b>assume </b><a NAME="E1:4_1_5"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E2:4_1_5"/><b>then </b>
<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">H</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font></span>)</span>))
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:4_1_5"/>
(  not <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>] or <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:4_1"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang.html#S1">ZF_LANG:sch 1</a>(, , , );<br/></i>
<b>hence </b><a NAME="E8:4_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>7</sub></font>())]
 ;<br/>


</div>
