<?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  <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</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> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> </span><br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <b>means </b>$1 <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> ;<br/>
<a NAME="E1:154"/><i><font color="Green" title="E59">A1</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c5">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c5">x2</font>] )</span><br/>



<a NAME="E1:154_1"/><i><font color="Green" title="E60">A2</font></i>: 
 ex <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</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> <font color="Olive" title="b1">y1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">y2</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T167" target="_self" title="ZF_LANG1:th.167">Th167</a></i>;<br/>
<a NAME="E2:154_1"/>
 ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</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> <font color="Olive" title="b1">z1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">z2</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T169" target="_self" title="ZF_LANG1:th.169">Th169</a></i>;<br/>
<b>hence </b><a NAME="E3:154_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c5">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c5">x2</font>] )
 <b>by </b><i><a class="txt" href="#E1:154_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</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>
<a NAME="E2:154"/><i><font color="Green" title="E60">A3</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c4">H</font>] )</span><br/>

<b>assume </b><a NAME="E1:154_2"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c4">H</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">F</font> = <font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:154_2"/>
 <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T170" target="_self" title="ZF_LANG1:th.170">Th170</a></i>;<br/>
<b>hence </b><a NAME="E4:154_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c4">H</font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</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>
<a NAME="E3:154"/><i><font color="Green" title="E61">A4</font></i>: 
for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">H2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">H1</font>, <font color="Maroon" title="c5">H2</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">H2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H2</font>] )</span><br/>



<b>assume </b><a NAME="E1:154_3"/>
( <font color="Maroon" title="c4">H1</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  &amp; <font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H2</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">F1</font> = <font color="Maroon" title="c4">H1</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>, <font color="Maroon" title="c7">F2</font> = <font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:154_3"/>
<font color="Maroon" title="c6">F1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="zf_lang1.html#E175"><i><font color="Green" title="E56">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:154_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">H2</font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</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>
<a NAME="E4:154"/><i><font color="Green" title="E62">A5</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">z</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Olive" title="b2">z</font>,<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="154_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">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">z</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Olive" title="b1">z</font>,<font color="Maroon" title="c4">H</font>]</span><br/><b>let </b><font color="Maroon" title="c5">z</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>] )</span><br/>



<b>assume </b><a NAME="E1:154_4"/>
<font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">F</font> = <font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> as   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</a></i>;<br/>
<a NAME="E3:154_4"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="E5:154_4"/><i><font color="Green" title="E63">A6</font></i>: 
( ( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font> ) or ( <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> ) )
 ;<br/>
<a NAME="E6:154_4"/>
 <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">s</font>,<font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font></span>)</span> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="#E5:154_4"><i><font color="Green" title="E63">A6</font></i></a>, <a class="txt" href="zf_lang1.html#E176"><i><font color="Green" title="E57">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:154_4"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>]
 <b>by </b><i><a class="ref" href="zf_lang.html#T14" title="ZF_LANG:th.14">ZF_LANG:14</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>
<a NAME="E5:154"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" target="_self" title="ZF_LANG1:sch.1">ZF_LANG1:sch 1</a>(<a class="txt" href="#E1:154"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="#E2:154"><i><font color="Green" title="E60">A3</font></i></a>, <a class="txt" href="#E3:154"><i><font color="Green" title="E61">A4</font></i></a>, <a class="txt" href="#E4:154"><i><font color="Green" title="E62">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:154"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K5" title="ZF_LANG1:func.5">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang.html#K9" title="ZF_LANG:func.9">WFF</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
