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<a NAME="E1:60_1"/><i><font color="Green">E65</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">P<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><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="E1:60_1_1"/><i><font color="Green">E66</font></i>: 
( <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="E2:60_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">H</font> <a href="zf_lang.html#NR1" target="_self">is_equality</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x1</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">x2</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><a NAME="E1:60_1_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="zf_lang.html#K4">'='</a> <font color="Maroon">x2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D10">ZF_LANG:def 10</a><br/>

<b>thus </b><a NAME="E2:60_1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60_1_1"/>
( <font color="Maroon">H</font> <a href="zf_lang.html#NR2" target="_self">is_membership</a>  implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">x1</font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">x2</font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><a NAME="E1:60_1_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="zf_lang.html#K5">'in'</a> <font color="Maroon">x2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D11">ZF_LANG:def 11</a><br/>

<b>thus </b><a NAME="E2:60_1_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:60_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:60_1"/><i><font color="Green">E70</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">P<sub>1</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_1_2"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V4">negative</a>
 ;<br/>

<a NAME="E2:60_1_2"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <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><a class="ref" href="zf_lang.html#D30">ZF_LANG:def 30</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_2"/>
(  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K20">the_argument_of</a> <font color="Maroon">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60_1"/><i><font color="Green">E71</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">P<sub>1</sub></font>[ <a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Olive">H</font>] &amp; <font color="Maroon">P<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">P<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_1_3"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V5">conjunctive</a>
 ;<br/>

<a NAME="E2:60_1_3"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21">the_left_argument_of</a> <font color="Maroon">H</font></span>)</span> <a href="zf_lang.html#K7">'&amp;'</a> <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><a class="ref" href="zf_lang.html#T58">ZF_LANG:58</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_3"/>
(  not <font color="Maroon">P<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">P<sub>1</sub></font>[ <a href="zf_lang.html#K22">the_right_argument_of</a> <font color="Maroon">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:60_1"/><i><font color="Green">E72</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">P<sub>1</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Olive">H</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_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="E1:60_1_4"/>
<font color="Maroon">H</font> is <a href="zf_lang.html#V6">universal</a>
 ;<br/>

<a NAME="E2:60_1_4"/><b>then </b>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">H</font></span>)</span>,<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><a class="ref" href="zf_lang.html#T62">ZF_LANG:62</a></i>;<br/>
<b>hence </b><a NAME="E3:60_1_4"/>
(  not <font color="Maroon">P<sub>1</sub></font>[ <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">H</font>] or <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">H</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E5:60_1"/>
for <font color="Olive">H</font> being  <a href="zf_lang.html#NM2">ZF-formula</a> holds  <font color="Maroon">P<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>(, , , );</i><br/>


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