<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st  not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b1">H</font> holds <br/>( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Olive" title="b1">H</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c3">H</font> implies ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> ) ) )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <span class="kw">means </span>(  not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> $1 implies ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> $1 implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default">$1 <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> $1</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> $1 implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default">$1 <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> $1 ) ) );<br/>
<a NAME="E1:2"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Olive" title="b2">H2</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H1</font>]
 <span class="kw">and </span><br/><a NAME="E2:2_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">H2</font>]
 <span class="kw">and </span><br/><a NAME="E3:2_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
 not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> ) )</span><br/>

<a NAME="E4:2_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T61" onmouseover="rs('zf_lang1/T61')" onmouseout="rh()">ZF_LANG1:61</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">H</font> = <font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font>;<br/>
<a NAME="E5:2_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T61" onmouseover="rs('zf_lang1/T61')" onmouseout="rh()">ZF_LANG1:61</a></span>;<br/>
<a NAME="E6:2_1"/><span class="lab"><font color="Green" title="E9">A7</font></span>: 
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T141" onmouseover="rs('zf_lang1/T141')" onmouseout="rh()">ZF_LANG1:141</a></span>;<br/>
<a NAME="E7:2_1"/><span class="lab"><font color="Green" title="E10">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</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_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T158" onmouseover="rs('zf_lang1/T158')" onmouseout="rh()">ZF_LANG1:158</a></span>;<br/>
<span class="kw">thus </span><a NAME="E8:2_1"/>
( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1"/><span class="lab"><font color="Green" title="E11">A9</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<div><span class="lab"><font color="Green" title="E12">A10</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_1"/><span class="lab"><font color="Green" title="E13">A11</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:2_1_1_1"/><span class="kw">then </span>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>;<br/><span class="kw">hence </span> <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E1:2_1_1_1"><span class="lab"><font color="Green" title="E13">A11</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="xboole_1.html#T42" onmouseover="rs('xboole_1/T42')" onmouseout="rh()">XBOOLE_1:42</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E13">A12</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_1_1_2"/><span class="lab"><font color="Green" title="E14">A13</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:2_1_1_2"/><span class="kw">then </span>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>;<br/><span class="kw">hence </span> <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_1"><span class="lab"><font color="Green" title="E11">A9</font></span></a>, <a class="txt" href="#E1:2_1_1_2"><span class="lab"><font color="Green" title="E14">A13</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="xboole_1.html#T42" onmouseover="rs('xboole_1/T42')" onmouseout="rh()">XBOOLE_1:42</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font> &amp; <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_1_1_3"/><span class="lab"><font color="Green" title="E14">A14</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font>
 <span class="kw">and </span><br/><a NAME="E2:2_1_1_3"/><span class="lab"><font color="Green" title="E15">A15</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">thus </span> <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="txt" href="#E7:2_1"><span class="lab"><font color="Green" title="E10">A8</font></span></a>, <a class="txt" href="#E1:2_1_1_3"><span class="lab"><font color="Green" title="E14">A14</font></span></a>, <a class="txt" href="#E2:2_1_1_3"><span class="lab"><font color="Green" title="E15">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H1</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="xboole_1.html#T42" onmouseover="rs('xboole_1/T42')" onmouseout="rh()">XBOOLE_1:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T4" onmouseover="rs('xboole_1/T4')" onmouseout="rh()">XBOOLE_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:2_1_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_1_1"><span class="lab"><font color="Green" title="E12">A10</font></span></a>, <a class="txt" href="#E3:2_1_1"><span class="lab"><font color="Green" title="E13">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E9:2_1"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E10:2_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c5">H2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E2:2_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E3:2_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a>, <a class="txt" href="#E5:2_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="txt" href="#E4:2_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="txt" href="#E6:2_1"><span class="lab"><font color="Green" title="E9">A7</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="zf_lang1.html#T158" onmouseover="rs('zf_lang1/T158')" onmouseout="rh()">ZF_LANG1:158</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:2"/><span class="lab"><font color="Green" title="E4">A16</font></span>: 
 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#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b2">z</font>,<font color="Olive" title="b1">H</font>)]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c4">H</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c5">z</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:2_2"/><span class="lab"><font color="Green" title="E5">A17</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">H</font>]
 <span class="kw">and </span><br/><a NAME="E2:2_2"/><span class="lab"><font color="Green" title="E6">A18</font></span>: 
 not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">G</font> = <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>);<br/>
<a NAME="E3:2_2"/><span class="lab"><font color="Green" title="E7">A19</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T62" onmouseover="rs('zf_lang1/T62')" onmouseout="rh()">ZF_LANG1:62</a></span>;<br/>
<a NAME="E4:2_2"/>
( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">s</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><br/><a NAME="E6:2_2"/><span class="lab"><font color="Green" title="E8">A20</font></span>: 
( ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> ) or ( <font color="Maroon" title="c5">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z</font> ) )
 ;<br/>
<a NAME="E7:2_2"/><span class="lab"><font color="Green" title="E9">A21</font></span>: 
<span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="ref" href="zf_lang1.html#T159" onmouseover="rs('zf_lang1/T159')" onmouseout="rh()">ZF_LANG1:159</a>, <a class="ref" href="zf_lang1.html#T160" onmouseover="rs('zf_lang1/T160')" onmouseout="rh()">ZF_LANG1:160</a></span>;<br/>
<a NAME="E8:2_2"/><span class="lab"><font color="Green" title="E10">A22</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">s</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">s</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T62" onmouseover="rs('zf_lang1/T62')" onmouseout="rh()">ZF_LANG1:62</a></span>;<br/>
<a NAME="E9:2_2"/><span class="lab"><font color="Green" title="E11">A23</font></span>: 
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T142" onmouseover="rs('zf_lang1/T142')" onmouseout="rh()">ZF_LANG1:142</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:2_2"/>
( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_2_1"/><span class="lab"><font color="Green" title="E12">A24</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A25</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:2_2_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_2_1_1"/><span class="lab"><font color="Green" title="E14">A26</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<a NAME="E2:2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E1:2_2_1"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E4:2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A27</font></span>: 
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E5:2_2_1_1"/>
 not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E2:2_2_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E1:2_2_1_1"><span class="lab"><font color="Green" title="E14">A26</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:2_2_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E4:2_2_1_1"><span class="lab"><font color="Green" title="E15">A27</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:2_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:2_2_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E4:2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<a NAME="E5:2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E6:2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A28</font></span>: 
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E7:2_2_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E8:2_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E9:2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A29</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E10:2_2_1"/>
 not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E6:2_2_1"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E1:2_2_1"><span class="lab"><font color="Green" title="E12">A24</font></span></a>, <a class="txt" href="#E2:2_2_1"><span class="lab"><font color="Green" title="E13">A25</font></span></a>, <a class="txt" href="#E9:2_2_1"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E11:2_2"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span><br/>

<a NAME="E12:2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A30</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font> or <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E13:2_2"/><span class="lab"><font color="Green" title="E13">A31</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T151" onmouseover="rs('zf_lang1/T151')" onmouseout="rh()">ZF_LANG1:151</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E14">A32</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_2_2"/><span class="lab"><font color="Green" title="E15">A33</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span><br/><span class="kw">thus </span><a NAME="E2:2_2_2"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">thus </span><a NAME="E1:2_2_2_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:2_2_2_1_1"/><span class="lab"><font color="Green" title="E16">A34</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span><br/>

<a NAME="E2:2_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A35</font></span>: 
 not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E12:2_2"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="txt" href="#E1:2_2_2"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E3:2_2_2_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E12:2_2"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="txt" href="#E1:2_2_2"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="txt" href="#E1:2_2_2_1_1"><span class="lab"><font color="Green" title="E16">A34</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:2_2_2_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E2:2_2_2_1_1"><span class="lab"><font color="Green" title="E17">A35</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:2_2_2_1"/><span class="lab"><font color="Green" title="E16">A36</font></span>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<a NAME="E3:2_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E13:2_2"><span class="lab"><font color="Green" title="E13">A31</font></span></a></span>;<br/>
<a NAME="E4:2_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A37</font></span>: 
 not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:2_2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="txt" href="#E12:2_2"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="txt" href="#E1:2_2_2"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="txt" href="#E2:2_2_2_1"><span class="lab"><font color="Green" title="E16">A36</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:2_2_2_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E12:2_2"><span class="lab"><font color="Green" title="E12">A30</font></span></a>, <a class="txt" href="#E1:2_2_2"><span class="lab"><font color="Green" title="E15">A33</font></span></a>, <a class="txt" href="#E4:2_2_2_1"><span class="lab"><font color="Green" title="E17">A37</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_2_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:2_2_3"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span></span><br/><a NAME="E2:2_2_3"/><span class="kw">then </span>
( (  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> ) or  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_2"><span class="lab"><font color="Green" title="E8">A20</font></span></a>, <a class="txt" href="#E1:2_2"><span class="lab"><font color="Green" title="E5">A17</font></span></a>, <a class="txt" href="#E2:2_2"><span class="lab"><font color="Green" title="E6">A18</font></span></a>, <a class="txt" href="#E8:2_2"><span class="lab"><font color="Green" title="E10">A22</font></span></a>, <a class="txt" href="#E9:2_2"><span class="lab"><font color="Green" title="E11">A23</font></span></a>, <a class="txt" href="#E7:2_2"><span class="lab"><font color="Green" title="E9">A21</font></span></a>, <a class="txt" href="#E13:2_2"><span class="lab"><font color="Green" title="E13">A31</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a>, <a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>;<br/><span class="kw">hence </span><a NAME="E3:2_2_3"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:2_2"><span class="lab"><font color="Green" title="E7">A19</font></span></a>, <a class="ref" href="zfmisc_1.html#T57" onmouseover="rs('zfmisc_1/T57')" onmouseout="rh()">ZFMISC_1:57</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:2_2"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c4">H</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:2_2"><span class="lab"><font color="Green" title="E14">A32</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:2"/><span class="lab"><font color="Green" title="E5">A38</font></span>: 
 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#K3" title="ZF_LANG:func.3">'='</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#K4" title="ZF_LANG:func.4">'in'</a> <font color="Olive" title="b2">x2</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</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#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font>] )</span><br/>



<a NAME="E1:2_3"/><span class="lab"><font color="Green" title="E6">A39</font></span>: 
( <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c5">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> )
 
;<br/>
<a NAME="E2:2_3"/>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><br/><a NAME="E4:2_3"/><span class="lab"><font color="Green" title="E7">A40</font></span>: 
( ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> ) or ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> ) or ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> ) or ( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c5">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_3"><span class="lab"><font color="Green" title="E6">A39</font></span></a></span>;<br/>
<a NAME="E5:2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_3"><span class="lab"><font color="Green" title="E7">A40</font></span></a>, <a class="ref" href="zf_lang1.html#T152" onmouseover="rs('zf_lang1/T152')" onmouseout="rh()">ZF_LANG1:152</a></span>;<br/>
<a NAME="E6:2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A41</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T58" onmouseover="rs('zf_lang1/T58')" onmouseout="rh()">ZF_LANG1:58</a></span>;<br/>
<a NAME="E7:2_3"/><span class="lab"><font color="Green" title="E9">A42</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T58" onmouseover="rs('zf_lang1/T58')" onmouseout="rh()">ZF_LANG1:58</a></span>;<br/>
<a NAME="E8:2_3"/><span class="lab"><font color="Green" title="E10">A43</font></span>: 
 <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T138" onmouseover="rs('zf_lang1/T138')" onmouseout="rh()">ZF_LANG1:138</a></span>;<br/>
<span class="kw">thus </span><a NAME="E9:2_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">'in'</a> <font color="Maroon" title="c5">x2</font>]</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_3_1"/>
 not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:2_3_1"/>
( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_3_1_1"/><span class="lab"><font color="Green" title="E11">A44</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">thus </span><a NAME="E2:2_3_1_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmodel2.html#T1" target="_self" onmouseover="rs('zfmodel2/T1')" onmouseout="rh()">Th1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:2_3_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<a NAME="E4:2_3_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:2_3_1_1"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:2_3_1_1"/><span class="kw">then </span>
( ( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> ) &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_3"><span class="lab"><font color="Green" title="E9">A42</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:2_3_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_3"><span class="lab"><font color="Green" title="E7">A40</font></span></a>, <a class="txt" href="#E6:2_3"><span class="lab"><font color="Green" title="E8">A41</font></span></a>, <a class="txt" href="#E7:2_3"><span class="lab"><font color="Green" title="E9">A42</font></span></a>, <a class="txt" href="#E1:2_3_1_1"><span class="lab"><font color="Green" title="E11">A44</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:2_3_1"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E4:2_3_1"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:2_3"><span class="lab"><font color="Green" title="E9">A42</font></span></a>, <a class="txt" href="#E8:2_3"><span class="lab"><font color="Green" title="E10">A43</font></span></a>, <a class="ref" href="zf_lang1.html#T182" onmouseover="rs('zf_lang1/T182')" onmouseout="rh()">ZF_LANG1:182</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E10:2_3"/><span class="lab"><font color="Green" title="E11">A45</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c5">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T59" onmouseover="rs('zf_lang1/T59')" onmouseout="rh()">ZF_LANG1:59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E11:2_3"/>
 not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> ) )</span><br/>

<a NAME="E12:2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_3"><span class="lab"><font color="Green" title="E7">A40</font></span></a>, <a class="ref" href="zf_lang1.html#T154" onmouseover="rs('zf_lang1/T154')" onmouseout="rh()">ZF_LANG1:154</a></span>;<br/>
<a NAME="E13:2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A46</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T59" onmouseover="rs('zf_lang1/T59')" onmouseout="rh()">ZF_LANG1:59</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:2_3"/>
( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_3_2"/><span class="lab"><font color="Green" title="E13">A47</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">thus </span><a NAME="E2:2_3_2"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmodel2.html#T1" target="_self" onmouseover="rs('zfmodel2/T1')" onmouseout="rh()">Th1</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:2_3_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span><br/>

<a NAME="E4:2_3_2"/><span class="kw">then </span>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:2_3_2"/><span class="kw">then </span>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> &amp;  not <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E6:2_3_2"/><span class="kw">then </span>
( ( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x2</font> ) &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> ) or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:2_3"><span class="lab"><font color="Green" title="E11">A45</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:2_3_2"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:2_3"><span class="lab"><font color="Green" title="E7">A40</font></span></a>, <a class="txt" href="#E13:2_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E10:2_3"><span class="lab"><font color="Green" title="E11">A45</font></span></a>, <a class="txt" href="#E1:2_3_2"><span class="lab"><font color="Green" title="E13">A47</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E15:2_3"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E16:2_3"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c5">x2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:2_3"><span class="lab"><font color="Green" title="E8">A41</font></span></a>, <a class="txt" href="#E7:2_3"><span class="lab"><font color="Green" title="E9">A42</font></span></a>, <a class="txt" href="#E13:2_3"><span class="lab"><font color="Green" title="E12">A46</font></span></a>, <a class="txt" href="#E10:2_3"><span class="lab"><font color="Green" title="E11">A45</font></span></a>, <a class="txt" href="#E8:2_3"><span class="lab"><font color="Green" title="E10">A43</font></span></a>, <a class="ref" href="zf_lang1.html#T182" onmouseover="rs('zf_lang1/T182')" onmouseout="rh()">ZF_LANG1:182</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:2"/><span class="lab"><font color="Green" title="E6">A48</font></span>: 
 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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c4">H</font>] )</span><br/>

<a NAME="E1:2_4"/><span class="lab"><font color="Green" title="E7">A49</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c4">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c4">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T60" onmouseover="rs('zf_lang1/T60')" onmouseout="rh()">ZF_LANG1:60</a></span>;<br/>
<a NAME="E2:2_4"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T60" onmouseover="rs('zf_lang1/T60')" onmouseout="rh()">ZF_LANG1:60</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:2_4"/>
( <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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c4">H</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:2_4"><span class="lab"><font color="Green" title="E7">A49</font></span></a>, <a class="ref" href="zf_lang1.html#T140" onmouseover="rs('zf_lang1/T140')" onmouseout="rh()">ZF_LANG1:140</a>, <a class="ref" href="zf_lang1.html#T156" onmouseover="rs('zf_lang1/T156')" onmouseout="rh()">ZF_LANG1:156</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:2"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang1.html#S1" onmouseover="rs('zf_lang1/S1')" onmouseout="rh()">ZF_LANG1:sch 1</a>(<a class="txt" href="#E3:2"><span class="lab"><font color="Green" title="E5">A38</font></span></a>, <a class="txt" href="#E4:2"><span class="lab"><font color="Green" title="E6">A48</font></span></a>, <a class="txt" href="#E1:2"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E2:2"><span class="lab"><font color="Green" title="E4">A16</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:2"/>
(  not <font color="Maroon" title="c2">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c3">H</font> implies ( ( <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) &amp; (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> implies  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c3">H</font> ) ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
