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

<span class="kw">let </span><font color="Maroon" title="c6">E</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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><br/>  for <font color="Olive" title="b2">f</font>, <font color="Olive" title="b3">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b4">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b2">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">x</font> holds <br/> not <font color="Olive" title="b4">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> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b3">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <span class="kw">means </span> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b3">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> holds <br/> not <font color="Olive" title="b3">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 ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1;<br/>
<a NAME="E1:16"/><span class="lab"><font color="Green" title="E11">A1</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="Olive" title="b1">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  holds <br/><font color="Maroon">S<sub>2</sub></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="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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" title="c7">H</font> is  <a href="zf_lang.html#V7" title="ZF_LANG:attr.7">atomic</a>  implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
( <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  or <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="zf_lang.html#D15" onmouseover="rs('zf_lang/D15')" onmouseout="rh()">ZF_LANG:def 15</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>2</sub></font>[<font color="Maroon" title="c7">H</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:16_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E3:16_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">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"> <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/>

<div><span class="lab"><font color="Green" title="E15">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_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"> ( <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</a>  implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_1"/><span class="lab"><font color="Green" title="E16">A6</font></span>: 
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V3" title="ZF_LANG:attr.3">being_membership</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" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A7</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font> <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"><span class="p2">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span></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_model.html#T1" onmouseover="rs('zf_model/T1')" onmouseout="rh()">ZF_MODEL:1</a></span>;<br/><a NAME="E3:16_1_1"/><span class="kw">then </span>
 <a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E4:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E5:16_1_1"/>
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E6:16_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E7:16_1_1"/><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="zf_lang.html#T37" onmouseover="rs('zf_lang/T37')" onmouseout="rh()">ZF_LANG:37</a></span>;<br/><a NAME="E8:16_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_1_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:16_1_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E6:16_1_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</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><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_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"> ( <font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>  implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:16_1_2"/><span class="lab"><font color="Green" title="E16">A11</font></span>: 
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</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" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/><a NAME="E2:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A12</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font> <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"><span class="p2">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span></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_model.html#T1" onmouseover="rs('zf_model/T1')" onmouseout="rh()">ZF_MODEL:1</a></span>;<br/><a NAME="E3:16_1_2"/><span class="kw">then </span>
 <a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E4:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E5:16_1_2"/>
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E6:16_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E7:16_1_2"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="zf_lang.html#T36" onmouseover="rs('zf_lang/T36')" onmouseout="rh()">ZF_LANG:36</a></span>;<br/><a NAME="E8:16_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K17" title="ZF_LANG:func.17">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:16_1_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_1_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E4:16_1_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E6:16_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</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="E6:16_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E4:16_1"><span class="lab"><font color="Green" title="E15">A5</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="E2:16"/><span class="lab"><font color="Green" title="E12">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>  st <font color="Olive" title="b1">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Olive" title="b1">H</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></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="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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" title="c7">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E13">A17</font></span>: 
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V5" title="ZF_LANG:attr.5">conjunctive</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">S<sub>2</sub></font>[ <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>] or  not <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<a NAME="E2:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A18</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T40" onmouseover="rs('zf_lang/T40')" onmouseout="rh()">ZF_LANG:40</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:16_2"/><span class="lab"><font color="Green" title="E15">A19</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b3">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> holds <br/> not <font color="Olive" title="b3">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#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font> holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E4:16_2"/><span class="lab"><font color="Green" title="E16">A20</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b3">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> holds <br/> not <font color="Olive" title="b3">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#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font> holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E5:16_2"/><span class="lab"><font color="Green" title="E17">A21</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E6:16_2"/><span class="lab"><font color="Green" title="E18">A22</font></span>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">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"> <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/>

<a NAME="E7:16_2"/><span class="lab"><font color="Green" title="E19">A23</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</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> <span class="p2">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</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"><a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="ref" href="zf_model.html#T1" onmouseover="rs('zf_model/T1')" onmouseout="rh()">ZF_MODEL:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A24</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2_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">  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</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" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> implies  not <font color="Maroon" title="c10">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#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:16_2_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</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">  not <font color="Maroon" title="c10">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#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:16_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_2_1"/>
 not <font color="Maroon" title="c10">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#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_2"><span class="lab"><font color="Green" title="E19">A23</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>
<div><span class="lab"><font color="Green" title="E21">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_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">  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</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" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> implies  not <font color="Maroon" title="c10">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#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:16_2_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</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">  not <font color="Maroon" title="c10">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#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:16_2_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c10">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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:16_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:16_2_2"/>
 not <font color="Maroon" title="c10">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#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_2"><span class="lab"><font color="Green" title="E19">A23</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>
<a NAME="E10:16_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E6:16_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E11:16_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A26</font></span>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K21" title="ZF_LANG:func.21">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E8:16_2"><span class="lab"><font color="Green" title="E20">A24</font></span></a></span>;<br/>
<a NAME="E12:16_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E6:16_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E13:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K20" title="ZF_LANG:func.20">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_2"><span class="lab"><font color="Green" title="E15">A19</font></span></a>, <a class="txt" href="#E9:16_2"><span class="lab"><font color="Green" title="E21">A25</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:16_2"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E11:16_2"><span class="lab"><font color="Green" title="E22">A26</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</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="E3:16"/><span class="lab"><font color="Green" title="E13">A27</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="Olive" title="b1">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></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="16_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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" title="c7">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_3"/><span class="lab"><font color="Green" title="E14">A28</font></span>: 
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V6" title="ZF_LANG:attr.6">universal</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">S<sub>2</sub></font>[ <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<a NAME="E2:16_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A29</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T44" onmouseover="rs('zf_lang/T44')" onmouseout="rh()">ZF_LANG:44</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_3"/><span class="lab"><font color="Green" title="E16">A30</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b3">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> holds <br/> not <font color="Olive" title="b3">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#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font> holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:16_3"/><span class="lab"><font color="Green" title="E17">A31</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E5:16_3"/><span class="lab"><font color="Green" title="E18">A32</font></span>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">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"> <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font></span><br/>

<a NAME="E6:16_3"/><span class="lab"><font color="Green" title="E19">A33</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</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> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">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"><span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="ref" href="zf_model.html#T1" onmouseover="rs('zf_model/T1')" onmouseout="rh()">ZF_MODEL:1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_3_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">  for <font color="Olive" title="b1">j</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b2">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">x</font> holds <br/> <a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x</font> ) holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">j</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c10">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:16_3_1"/><span class="lab"><font color="Green" title="E20">A34</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</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" title="c6">E</font>,<font color="Maroon" title="c10">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span><br/><span class="kw">set </span><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>);<br/><div><span class="lab"><font color="Green" title="E21">A35</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_3_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">  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">x</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> implies  not <font color="Maroon" title="c12">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#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:16_3_1_1"/><span class="lab"><font color="Green" title="E22">A36</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</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">  not <font color="Maroon" title="c12">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#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:16_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A37</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/><a NAME="E3:16_3_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> &amp; <font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_3_1"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>;<br/><a NAME="E4:16_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A38</font></span>: 
 not <font color="Maroon" title="c12">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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16_3"><span class="lab"><font color="Green" title="E17">A31</font></span></a>, <a class="txt" href="#E1:16_3_1_1"><span class="lab"><font color="Green" title="E22">A36</font></span></a></span>;<br/><a NAME="E5:16_3_1_1"/>
 not <font color="Maroon" title="c12">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"><span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3_1_1"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:16_3_1_1"/>
 not <font color="Maroon" title="c12">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#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_3"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="txt" href="#E4:16_3_1_1"><span class="lab"><font color="Green" title="E24">A38</font></span></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><a NAME="E3:16_3_1"/>
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>;<br/><a NAME="E4:16_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="funct_7.html#K14" title="FUNCT_7:func.14">+*</a> (<span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span></span>)</span>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="txt" href="#E5:16_3"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="zf_model.html#T16" onmouseover="rs('zf_model/T16')" onmouseout="rh()">ZF_MODEL:16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:16_3_1"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c10">j</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K23" title="ZF_LANG:func.23">the_scope_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_3"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E2:16_3_1"><span class="lab"><font color="Green" title="E21">A35</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">hence </span><a NAME="E8:16_3"/>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_3"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="ref" href="zf_model.html#T16" onmouseover="rs('zf_model/T16')" onmouseout="rh()">ZF_MODEL:16</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="E4:16"/><span class="lab"><font color="Green" title="E14">A39</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="Olive" title="b1">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>2</sub></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="16_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">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" title="c7">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</a>  &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_4"/><span class="lab"><font color="Green" title="E15">A40</font></span>: 
<font color="Maroon" title="c7">H</font> is  <a href="zf_lang.html#V4" title="ZF_LANG:attr.4">negative</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">S<sub>2</sub></font>[ <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font>] or <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>] )</span><br/>

<a NAME="E2:16_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A41</font></span>: 
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c7">H</font> <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#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T1" onmouseover="rs('zf_model/T1')" onmouseout="rh()">ZF_MODEL:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:16_4"/><span class="lab"><font color="Green" title="E17">A42</font></span>: 
 for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</font>  st (  for <font color="Olive" title="b3">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Olive" title="b1">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">x</font> holds <br/> not <font color="Olive" title="b3">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#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Olive" title="b1">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font> holds <br/><font color="Maroon" title="c6">E</font>,<font color="Olive" title="b2">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">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"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">H</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">f</font>, <font color="Maroon" title="c9">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="zf_lang.html#K1" title="ZF_LANG:func.1">VAR</a>,<font color="Maroon" title="c6">E</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"> ( (  for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font> ) &amp; <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> implies <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:16_4"/><span class="lab"><font color="Green" title="E18">A43</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/> not <font color="Olive" title="b1">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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E5:16_4"/><span class="lab"><font color="Green" title="E19">A44</font></span>: 
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E6:16_4"/><span class="lab"><font color="Green" title="E20">A45</font></span>: 
 not <font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">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"> contradiction</span><br/>

<a NAME="E7:16_4"/><span class="lab"><font color="Green" title="E21">A46</font></span>: 
<font color="Maroon" title="c7">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16_4"><span class="lab"><font color="Green" title="E15">A40</font></span></a>, <a class="ref" href="zf_lang.html#D30" onmouseover="rs('zf_lang/D30')" onmouseout="rh()">ZF_LANG:def 30</a></span>;<br/>
<a NAME="E8:16_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c9">g</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16_4"><span class="lab"><font color="Green" title="E20">A45</font></span></a>, <a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/>
<a NAME="E9:16_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">E</font>,<font color="Maroon" title="c8">f</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K19" title="ZF_LANG:func.19">the_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16_4"><span class="lab"><font color="Green" title="E16">A41</font></span></a>, <a class="txt" href="#E3:16_4"><span class="lab"><font color="Green" title="E17">A42</font></span></a>, <a class="txt" href="#E4:16_4"><span class="lab"><font color="Green" title="E18">A43</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:16_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16_4"><span class="lab"><font color="Green" title="E21">A46</font></span></a>, <a class="txt" href="#E5:16_4"><span class="lab"><font color="Green" title="E19">A44</font></span></a>, <a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</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">thus </span><a NAME="E5:16"/>
 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>2</sub></font>[<font color="Olive" title="b1">H</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang.html#S1" onmouseover="rs('zf_lang/S1')" onmouseout="rh()">ZF_LANG:sch 1</a>(<a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E14">A39</font></span></a>, <a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E13">A27</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>
