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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="zf_lang.html#NM2">ZF-formula</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="zf_lang.html#NM1">Variable</a>;<br/>





<b>thus </b><a NAME="E1:177"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V11">existential</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="zf_lang.html#V11">existential</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="177_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">c<sub>5</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><a NAME="E1:177_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D23">ZF_LANG:def 23</a><br/>

<a NAME="E2:177_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E4:177_1"/><i><font color="Green">E77</font></i>: 
( ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) )
 ;<br/>
<a NAME="E5:177_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E1:177_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="zf_lang1.html#E4:177_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zf_lang1.html#T178" target="_self">Th178</a>, <a class="ref" href="zf_lang1.html#T179" target="_self">Th179</a></i>;<br/>
<b>hence </b><a NAME="E6:177_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="zf_lang.html#V11">existential</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22">ZF_LANG:22</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="zf_lang.html#NM1">Variable</a>, <font color="Maroon">c<sub>5</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><a NAME="E2:177"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K13">Ex</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D23">ZF_LANG:def 23</a><br/>

<a NAME="E3:177"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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#K8">All</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:177"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E4:177"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="zf_lang.html#V4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E5:177"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self">Th182</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="zf_lang.html#NM2">ZF-formula</a><b> such that </b><br/><a NAME="E7:177"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E8:177"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:177"><i><font color="Green">E77</font></i></a>, <a class="txt" href="zf_lang1.html#E7:177"><i><font color="Green">E78</font></i></a>, <a class="ref" href="zf_lang1.html#T170" target="_self">Th170</a></i>;<br/>
<a NAME="E9:177"/>
(  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> or  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="zf_lang.html#NM1">Variable</a><b> such that </b><br/><a NAME="E11:177"/><i><font color="Green">E80</font></i>: 
( (  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> ) or (  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font> ) )
 ;<br/>
<a NAME="E12:177"/>
<font color="Maroon">c<sub>6</sub></font> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="zf_lang.html#V6">universal</a>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E8:177"><i><font color="Green">E79</font></i></a>, <a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E13:177"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is <a href="zf_lang.html#V6">universal</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T184" target="_self">Th184</a></i>;<br/>
<a NAME="E14:177"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T62">ZF_LANG:62</a></i>;<br/>
<a NAME="E15:177"/><b>then </b>
 <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K8">All</a> <font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E8:177"><i><font color="Green">E79</font></i></a>, <a class="txt" href="zf_lang1.html#E11:177"><i><font color="Green">E80</font></i></a>, <a class="ref" href="zf_lang1.html#T173" target="_self">Th173</a>, <a class="ref" href="zf_lang1.html#T174" target="_self">Th174</a></i>;<br/>
<a NAME="E16:177"/><b>then </b>
 <a href="zf_lang.html#K6">'not'</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T12">ZF_LANG:12</a></i>;<br/>
<a NAME="E17:177"/><b>then </b>
<span class="p1">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="zf_lang1.html#K6" target="_self">/</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="zf_lang.html#V4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16">ZF_LANG:16</a></i>;<br/>
<a NAME="E18:177"/><b>then </b>
 <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font> is <a href="zf_lang.html#V4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self">Th182</a></i>;<br/>
<a NAME="E19:177"/><b>then </b>
 <a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></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> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#D30">ZF_LANG:def 30</a></i>;<br/>
<a NAME="E20:177"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="zf_lang.html#K13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23">bound_in</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K20">the_argument_of</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24">the_scope_of</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E7:177"><i><font color="Green">E78</font></i></a>, <a class="txt" href="zf_lang1.html#E14:177"><i><font color="Green">E81</font></i></a></i>;<br/>
<b>hence </b><a NAME="E21:177"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="zf_lang.html#V11">existential</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22">ZF_LANG:22</a></i>;<br/>


</div>
