<?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#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="hidden.html#R2" title="HIDDEN: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="hidden.html#R2" title="HIDDEN: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="E1:17"/><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" onclick="hs2(this)" href="javascript:()" title="17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E1:17_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="Red">:: according to </font></span><a class="ref" href="zf_lang.html#D15" title="ZF_LANG:def.15">ZF_LANG:def 15</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="E2:17_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="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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E3:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs2(this)" href="javascript:()" title="17_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:17_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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">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" title="ZF_MODEL:th.1">ZF_MODEL:1</a></span>;<br/><a NAME="E3:17_1_1"/><span class="kw">then </span>
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E4:17_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#K18" title="ZF_LANG:func.18">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E5:17_1_1"/>
 <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E6:17_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#K19" title="ZF_LANG:func.19">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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E7:17_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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="ref" href="zf_lang.html#T54" title="ZF_LANG:th.54">ZF_LANG:54</a></span>;<br/><a NAME="E8:17_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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN: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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>;<br/><span class="kw">hence </span><a NAME="E9:17_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="#E7:17_1_1"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="txt" href="#E4:17_1_1"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="txt" href="#E6:17_1_1"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:17_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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p2">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">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" title="ZF_MODEL:th.1">ZF_MODEL:1</a></span>;<br/><a NAME="E3:17_1_2"/><span class="kw">then </span>
 <a href="zf_lang.html#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E4:17_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#K18" title="ZF_LANG:func.18">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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E5:17_1_2"/>
 <a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1_2"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E6:17_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#K19" title="ZF_LANG:func.19">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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/><a NAME="E7:17_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#K18" title="ZF_LANG:func.18">Var1</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_1_2"><span class="lab"><font color="Green" title="E16">A11</font></span></a>, <a class="ref" href="zf_lang.html#T53" title="ZF_LANG:th.53">ZF_LANG:53</a></span>;<br/><a NAME="E8:17_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#K18" title="ZF_LANG:func.18">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#K19" title="ZF_LANG:func.19">Var2</a> <font color="Maroon" title="c7">H</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>;<br/><span class="kw">hence </span><a NAME="E9:17_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="#E7:17_1_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E4:17_1_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E6:17_1_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:17_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="#E1:17_1"><span class="lab"><font color="Green" title="E12">A2</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:17"/><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#K21" title="ZF_LANG:func.21">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#K22" title="ZF_LANG:func.22">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" onclick="hs2(this)" href="javascript:()" title="17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K21" title="ZF_LANG:func.21">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#K22" title="ZF_LANG:func.22">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="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K21" title="ZF_LANG:func.21">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#K22" title="ZF_LANG:func.22">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="E2:17_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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K22" title="ZF_LANG:func.22">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#T58" title="ZF_LANG:th.58">ZF_LANG:58</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:17_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="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#K21" title="ZF_LANG:func.21">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#K21" title="ZF_LANG:func.21">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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E4:17_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="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#K22" title="ZF_LANG:func.22">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#K22" title="ZF_LANG:func.22">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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="E5:17_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="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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E6:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E7:17_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#K21" title="ZF_LANG:func.21">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#K22" title="ZF_LANG:func.22">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="#E1:17_2"><span class="lab"><font color="Green" title="E13">A17</font></span></a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c10">x</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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:17_2_1"/><span class="kw">then </span>
not <font color="Maroon" title="c10">x</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_2_1"/>
not <font color="Maroon" title="c10">x</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#K22" title="ZF_LANG:func.22">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="#E7:17_2"><span class="lab"><font color="Green" title="E19">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs2(this)" href="javascript:()" title="17_2_2"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c10">x</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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:17_2_2"/><span class="kw">then </span>
not <font color="Maroon" title="c10">x</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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E17">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:17_2_2"/>
not <font color="Maroon" title="c10">x</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#K21" title="ZF_LANG:func.21">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="#E7:17_2"><span class="lab"><font color="Green" title="E19">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:17_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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E6:17_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E11:17_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#K22" title="ZF_LANG:func.22">the_right_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_2"><span class="lab"><font color="Green" title="E16">A20</font></span></a>, <a class="txt" href="#E8:17_2"><span class="lab"><font color="Green" title="E20">A24</font></span></a></span>;<br/>
<a NAME="E12:17_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_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E6:17_2"><span class="lab"><font color="Green" title="E18">A22</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>;<br/>
<a NAME="E13:17_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#K21" title="ZF_LANG:func.21">the_left_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_2"><span class="lab"><font color="Green" title="E15">A19</font></span></a>, <a class="txt" href="#E9:17_2"><span class="lab"><font color="Green" title="E21">A25</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:17_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="#E2:17_2"><span class="lab"><font color="Green" title="E14">A18</font></span></a>, <a class="txt" href="#E11:17_2"><span class="lab"><font color="Green" title="E22">A26</font></span></a>, <a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:17"/><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#K24" title="ZF_LANG:func.24">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" onclick="hs2(this)" href="javascript:()" title="17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K24" title="ZF_LANG:func.24">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="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon">S<sub>2</sub></font>[ <a href="zf_lang.html#K24" title="ZF_LANG:func.24">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="E2:17_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#K8" title="ZF_LANG:func.8">All</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="c7">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="zf_lang.html#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T62" title="ZF_LANG:th.62">ZF_LANG:62</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:17_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="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#K24" title="ZF_LANG:func.24">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#K24" title="ZF_LANG:func.24">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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="E4:17_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="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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E5:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E6:17_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#K24" title="ZF_LANG:func.24">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#K23" title="ZF_LANG:func.23">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="#E1:17_3"><span class="lab"><font color="Green" title="E14">A28</font></span></a>, <a class="ref" href="zf_model.html#T1" title="ZF_MODEL:th.1">ZF_MODEL:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_3_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K23" title="ZF_LANG:func.23">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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_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#K23" title="ZF_LANG:func.23">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K24" title="ZF_LANG:func.24">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#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="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#K23" title="ZF_LANG:func.23">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" onclick="hs2(this)" href="javascript:()" title="17_3_1_1"><span class="kw">now </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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K15" title="FUNCT_7:func.15">+*</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">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#K23" title="ZF_LANG:func.23">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="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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:17_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#K15" title="FUNCT_7:func.15">+*</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">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#K23" title="ZF_LANG:func.23">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c12">x</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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font></span>)</span></span><br/><a NAME="E2:17_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#K23" title="ZF_LANG:func.23">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#T130" title="FUNCT_7:th.130">FUNCT_7:130</a></span>;<br/><a NAME="E3:17_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#K15" title="FUNCT_7:func.15">+*</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">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#K23" title="ZF_LANG:func.23">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="#E1:17_3_1"><span class="lab"><font color="Green" title="E20">A34</font></span></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>;<br/><a NAME="E4:17_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="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="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_3"><span class="lab"><font color="Green" title="E17">A31</font></span></a>, <a class="txt" href="#E1:17_3_1_1"><span class="lab"><font color="Green" title="E22">A36</font></span></a></span>;<br/><a NAME="E5:17_3_1_1"/>
not <font color="Maroon" title="c12">x</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"><span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">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="#E2:17_3_1_1"><span class="lab"><font color="Green" title="E23">A37</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:17_3_1_1"/>
not <font color="Maroon" title="c12">x</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#K24" title="ZF_LANG:func.24">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="#E6:17_3"><span class="lab"><font color="Green" title="E19">A33</font></span></a>, <a class="txt" href="#E4:17_3_1_1"><span class="lab"><font color="Green" title="E24">A38</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E3:17_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#K15" title="FUNCT_7:func.15">+*</a> <span class="p2">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">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#K23" title="ZF_LANG:func.23">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#K23" title="ZF_LANG:func.23">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#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></span>;<br/><a NAME="E4:17_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#K15" title="FUNCT_7:func.15">+*</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K23" title="ZF_LANG:func.23">bound_in</a> <font color="Maroon" title="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#K23" title="ZF_LANG:func.23">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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_3"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E18">A32</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>;<br/><span class="kw">hence </span><a NAME="E5:17_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#K24" title="ZF_LANG:func.24">the_scope_of</a> <font color="Maroon" title="c7">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E16">A30</font></span></a>, <a class="txt" href="#E2:17_3_1"><span class="lab"><font color="Green" title="E21">A35</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:17_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="#E2:17_3"><span class="lab"><font color="Green" title="E15">A29</font></span></a>, <a class="ref" href="zf_model.html#T16" title="ZF_MODEL:th.16">ZF_MODEL:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:17"/><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#K20" title="ZF_LANG:func.20">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" onclick="hs2(this)" href="javascript:()" title="17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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#K20" title="ZF_LANG:func.20">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="E1:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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_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="E2:17_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#K20" title="ZF_LANG:func.20">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" title="ZF_MODEL:th.1">ZF_MODEL:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:17_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="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#K20" title="ZF_LANG:func.20">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#K20" title="ZF_LANG:func.20">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#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c7">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="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="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="E4:17_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="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="c7">H</font>
 <span class="kw">and </span><br/><a NAME="E5:17_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="E6:17_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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:17_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#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K20" title="ZF_LANG:func.20">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="#E1:17_4"><span class="lab"><font color="Green" title="E15">A40</font></span></a>, <a class="ref" href="zf_lang.html#D30" title="ZF_LANG:def.30">ZF_LANG:def 30</a></span>;<br/>
<a NAME="E8:17_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#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17_4"><span class="lab"><font color="Green" title="E20">A45</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>;<br/>
<a NAME="E9:17_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#K20" title="ZF_LANG:func.20">the_argument_of</a> <font color="Maroon" title="c7">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_4"><span class="lab"><font color="Green" title="E16">A41</font></span></a>, <a class="txt" href="#E3:17_4"><span class="lab"><font color="Green" title="E17">A42</font></span></a>, <a class="txt" href="#E4:17_4"><span class="lab"><font color="Green" title="E18">A43</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:17_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:17_4"><span class="lab"><font color="Green" title="E21">A46</font></span></a>, <a class="txt" href="#E5:17_4"><span class="lab"><font color="Green" title="E19">A44</font></span></a>, <a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E5:17"/>
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" title="ZF_LANG:sch.1">ZF_LANG:sch 1</a>(<a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E4:17"><span class="lab"><font color="Green" title="E14">A39</font></span></a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E12">A16</font></span></a>, <a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E13">A27</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
