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

<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> iff <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> iff <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> )</span><br/>





<b>thus </b><a NAME="E1:176"/>
( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> implies <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> implies <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="176_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">z</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>, <font color="Maroon" title="c5">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><a NAME="E2:176_1"/><i><font color="Green" title="E76">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">H1</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D23" title="ZF_LANG:def.23">ZF_LANG:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a></span><br/>

<a NAME="E3:176_1"/>
( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c6">s</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><br/><a NAME="E5:176_1"/><i><font color="Green" title="E77">A2</font></i>: 
( ( <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> ) or ( <font color="Maroon" title="c4">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> ) )
 ;<br/>
<a NAME="E6:176_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c6">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E2:176_1"><i><font color="Green" title="E76">A1</font></i></a>, <a class="txt" href="zf_lang1.html#E5:176_1"><i><font color="Green" title="E77">A2</font></i></a>, <a class="ref" href="zf_lang1.html#T178" target="_self" title="ZF_LANG1:th.178">Th178</a>, <a class="ref" href="zf_lang1.html#T179" target="_self" title="ZF_LANG1:th.179">Th179</a></i>;<br/>
<b>hence </b><a NAME="E7:176_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>given </b><font color="Maroon" title="c4">z</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>, <font color="Maroon" title="c5">G</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><a NAME="E3:176"/><i><font color="Green" title="E76">A3</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <font color="Maroon" title="c4">z</font>,<font color="Maroon" title="c5">G</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D23" title="ZF_LANG:def.23">ZF_LANG:def 23</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a></span><br/>

<a NAME="E4:176"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:176"><i><font color="Green" title="E76">A3</font></i></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E5:176"/><b>then </b>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">H1</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><b> such that </b><br/><a NAME="E7:176"/><i><font color="Green" title="E77">A4</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H1</font>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E8:176"/><i><font color="Green" title="E78">A5</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E3:176"><i><font color="Green" title="E76">A3</font></i></a>, <a class="txt" href="zf_lang1.html#E7:176"><i><font color="Green" title="E77">A4</font></i></a>, <a class="ref" href="zf_lang1.html#T170" target="_self" title="ZF_LANG1:th.170">Th170</a></i>;<br/>
<a NAME="E9:176"/>
(  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> or  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c7">s</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><br/><a NAME="E11:176"/><i><font color="Green" title="E79">A6</font></i>: 
( (  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> ) or (  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font> ) )
 ;<br/>
<a NAME="E12:176"/>
<font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E8:176"><i><font color="Green" title="E78">A5</font></i></a>, <a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E13:176"/><b>then </b>
<font color="Maroon" title="c6">H1</font> is <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T184" target="_self" title="ZF_LANG1:th.184">Th184</a></i>;<br/>
<a NAME="E14:176"/><b>then </b><i><font color="Green" title="E80">A7</font></i>: 
<font color="Maroon" title="c6">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></i>;<br/>
<a NAME="E15:176"/><b>then </b>
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c4">z</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E8:176"><i><font color="Green" title="E78">A5</font></i></a>, <a class="txt" href="zf_lang1.html#E11:176"><i><font color="Green" title="E79">A6</font></i></a>, <a class="ref" href="zf_lang1.html#T173" target="_self" title="ZF_LANG1:th.173">Th173</a>, <a class="ref" href="zf_lang1.html#T174" target="_self" title="ZF_LANG1:th.174">Th174</a></i>;<br/>
<a NAME="E16:176"/><b>then </b>
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T12" title="ZF_LANG:th.12">ZF_LANG:12</a></i>;<br/>
<a NAME="E17:176"/><b>then </b>
<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</a></i>;<br/>
<a NAME="E18:176"/><b>then </b>
 <a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font> is <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T182" target="_self" title="ZF_LANG1:th.182">Th182</a></i>;<br/>
<a NAME="E19:176"/><b>then </b>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K13" title="ZF_LANG:func.13">Ex</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c6">H1</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_argument_of</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c6">H1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E7:176"><i><font color="Green" title="E77">A4</font></i></a>, <a class="txt" href="zf_lang1.html#E14:176"><i><font color="Green" title="E80">A7</font></i></a>, <a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></i>;<br/>
<b>hence </b><a NAME="E20:176"/>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V11" title="ZF_LANG:attr.11">existential</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T22" title="ZF_LANG:th.22">ZF_LANG:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
