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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being 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> <br/> for <font color="Olive" title="b2">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b3">v</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="Olive" title="b1">M</font>  st not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b2">H</font> holds <br/>( <font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Olive" title="b1">M</font>,<font color="Olive" title="b3">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b3">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b2">H</font> )</span><br/><b>let </b><font color="Maroon" title="c3">M</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:()"><i><font color="Red">::  thesis: </font></i></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">v</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="c3">M</font>  st not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b1">H</font> holds <br/>( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b2">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Olive" title="b1">H</font> )</span><br/><b>let </b><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font>  st not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> holds <br/>( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> )</span><br/><b>let </b><font color="Maroon" title="c5">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> implies ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )</span><br/>









<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <b>means </b>( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> $1 implies for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> $1 ) );<br/>
<a NAME="E1:13"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Olive" title="b2">x2</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font>] )</span><br/>



<a NAME="E1:13_1"/><i><font color="Green" title="E15">A2</font></i>: 
( ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> ) &amp; ( <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> ) )
 
;<br/>
<b>thus </b><a NAME="E2:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font>]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<b>let </b><font color="Maroon" title="c8">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><br/><a NAME="E3:13_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
( ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E4:13_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E3:13_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="zf_lang1.html#T166" title="ZF_LANG1:th.166">ZF_LANG1:166</a></i>;<br/>
<a NAME="E5:13_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y2</font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E3:13_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E6:13_1_1"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span><br/>

<a NAME="E2:13_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E4:13_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E5:13_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E7:13_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E8:13_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>;<br/>
<b>hence </b><a NAME="E9:13_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E4:13_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="zfmodel2.html#E5:13_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="zf_model.html#T12" title="ZF_MODEL:th.12">ZF_MODEL:12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:13_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<b>let </b><font color="Maroon" title="c8">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><br/><a NAME="E5:13_1"/><i><font color="Green" title="E16">A6</font></i>: 
( ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) or ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E6:13_1"/><i><font color="Green" title="E17">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E5:13_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="zf_lang1.html#T168" title="ZF_LANG1:th.168">ZF_LANG1:168</a></i>;<br/>
<a NAME="E7:13_1"/><i><font color="Green" title="E18">A8</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y2</font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E5:13_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/>
<b>thus </b><a NAME="E8:13_1"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span><br/>

<a NAME="E2:13_1_2"/><b>then </b>
<font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E6:13_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E7:13_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E9:13_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E10:13_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x2</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>;<br/>
<b>hence </b><a NAME="E11:13_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E6:13_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="txt" href="zfmodel2.html#E7:13_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="ref" href="zf_model.html#T13" title="ZF_MODEL:th.13">ZF_MODEL:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E2:13"/><i><font color="Green" title="E15">A9</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font>] )</span><br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E16">A10</font></i>: 
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H</font> implies for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font>]</span><br/>

<b>assume </b><a NAME="E2:13_2"/><i><font color="Green" title="E17">A11</font></i>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font> )</span><br/>

<b>let </b><font color="Maroon" title="c7">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font> )</span><br/>

<b>thus </b><a NAME="E3:13_2"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span><br/>

<a NAME="E2:13_2_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T170" title="ZF_LANG1:th.170">ZF_LANG1:170</a></i>;<br/>
<a NAME="E3:13_2_1"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E4:13_2_1"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_2"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="zfmodel2.html#E2:13_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="zf_lang1.html#T153" title="ZF_LANG1:th.153">ZF_LANG1:153</a></i>;<br/>
<b>hence </b><a NAME="E5:13_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:13_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E5:13_2"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<a NAME="E6:13_2"/><b>then </b>
not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_2"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="zfmodel2.html#E2:13_2"><i><font color="Green" title="E17">A11</font></i></a>, <a class="ref" href="zf_lang1.html#T153" title="ZF_LANG1:th.153">ZF_LANG1:153</a></i>;<br/>
<a NAME="E7:13_2"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_model.html#T14" title="ZF_MODEL:th.14">ZF_MODEL:14</a></i>;<br/>
<b>hence </b><a NAME="E8:13_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K6" title="ZF_LANG:func.6">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T170" title="ZF_LANG1:th.170">ZF_LANG1:170</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:13"/><i><font color="Green" title="E16">A12</font></i>: 
for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">H2</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">H1</font>, <font color="Maroon" title="c7">H2</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">H2</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13_3"/><i><font color="Green" title="E17">A13</font></i>: 
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H1</font> implies for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> ) )
 <b>and </b><br/><a NAME="E2:13_3"/><i><font color="Green" title="E18">A14</font></i>: 
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c7">H2</font> implies for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font>]</span><br/>

<b>assume </b><a NAME="E3:13_3"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )</span><br/>

<a NAME="E4:13_3"/><b>then </b><i><font color="Green" title="E19">A15</font></i>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c7">H2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T154" title="ZF_LANG1:th.154">ZF_LANG1:154</a></i>;<br/>
<b>let </b><font color="Maroon" title="c8">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )</span><br/>

<b>thus </b><a NAME="E5:13_3"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span><br/>

<a NAME="E2:13_3_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T172" title="ZF_LANG1:th.172">ZF_LANG1:172</a></i>;<br/>
<a NAME="E3:13_3_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E4:13_3_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_3"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="zfmodel2.html#E2:13_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="zfmodel2.html#E4:13_3"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:13_3_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font>
 <b>by </b><i><a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E6:13_3"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<a NAME="E7:13_3"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> )
 
<b>by </b><i><a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<a NAME="E8:13_3"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c7">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_3"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="zfmodel2.html#E2:13_3"><i><font color="Green" title="E18">A14</font></i></a>, <a class="txt" href="zfmodel2.html#E4:13_3"><i><font color="Green" title="E19">A15</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E9:13_3"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">H2</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_model.html#T15" title="ZF_MODEL:th.15">ZF_MODEL:15</a></i>;<br/>
<b>hence </b><a NAME="E10:13_3"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K7" title="ZF_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T172" title="ZF_LANG1:th.172">ZF_LANG1:172</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:13"/><i><font color="Green" title="E17">A16</font></i>: 
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/> for <font color="Olive" title="b2">z</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Olive" title="b2">z</font>,<font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Olive" title="b1">z</font>,<font color="Maroon" title="c6">H</font>]</span><br/><b>let </b><font color="Maroon" title="c7">z</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>] )</span><br/>



<b>assume </b><a NAME="E1:13_4"/><i><font color="Green" title="E18">A17</font></i>: 
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H</font> implies for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>]</span><br/>

<b>assume </b><a NAME="E2:13_4"/><i><font color="Green" title="E19">A18</font></i>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font> )</span><br/>

<a NAME="E3:13_4"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#T164" title="ZF_LANG1:th.164">ZF_LANG1:164</a></i>;<br/>
<a NAME="E4:13_4"/><b>then </b><i><font color="Green" title="E20">A19</font></i>: 
not <font color="Maroon" title="c1">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#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E2:13_4"><i><font color="Green" title="E19">A18</font></i></a></i>;<br/>
<a NAME="E5:13_4"/><i><font color="Green" title="E21">A20</font></i>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E2:13_4"><i><font color="Green" title="E19">A18</font></i></a>, <a class="ref" href="zf_lang1.html#T155" title="ZF_LANG1:th.155">ZF_LANG1:155</a></i>;<br/>
<a NAME="E6:13_4"/><b>then </b>
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H</font> &amp; not <font color="Maroon" title="c1">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"><font color="Maroon" title="c7">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c6">H</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="zf_lang1.html#T164" title="ZF_LANG1:th.164">ZF_LANG1:164</a></i>;<br/>
<a NAME="E7:13_4"/><b>then </b><i><font color="Green" title="E22">A21</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">z</font> &amp; not <font color="Maroon" title="c1">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="c6">H</font> &amp; ( for <font color="Olive" title="b1">v</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="c3">M</font> holds <br/> ( <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> ) ) )
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4"><i><font color="Green" title="E18">A17</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:13_4"/>
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> or <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> )
 
;<br/>
<b>then consider </b><font color="Maroon" title="c8">s</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><br/><a NAME="E10:13_4"/><i><font color="Green" title="E23">A22</font></i>: 
( ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) or ( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> ) )
 ;<br/>
<b>let </b><font color="Maroon" title="c9">v</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="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font> )</span><br/>

<b>thus </b><a NAME="E11:13_4"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_4_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span><br/>

<a NAME="E2:13_4_1"/><b>then </b><i><font color="Green" title="E24">A23</font></i>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c8">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="zfmodel2.html#E10:13_4"><i><font color="Green" title="E23">A22</font></i></a>, <a class="ref" href="zf_lang1.html#T173" title="ZF_LANG1:th.173">ZF_LANG1:173</a>, <a class="ref" href="zf_lang1.html#T174" title="ZF_LANG1:th.174">ZF_LANG1:174</a></i>;<br/>
<div><i><font color="Green" title="E25">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_1_1"/><i><font color="Green" title="E26">A25</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font></span><br/><a NAME="E1:13_4_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E2:13_4_1"><i><font color="Green" title="E24">A23</font></i></a>, <a class="txt" href="zfmodel2.html#E1:13_4_1_1"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><a NAME="E2:13_4_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E7:13_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="zfmodel2.html#E1:13_4_1_1"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="zf_lang1.html#T79" title="ZF_LANG1:th.79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><a NAME="E3:13_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E7:13_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></i>;<br/><a NAME="E4:13_4_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> &amp; <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T78" title="ZF_LANG1:th.78">ZF_LANG1:78</a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><b>hence </b><a NAME="E5:13_4_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T78" title="ZF_LANG1:th.78">ZF_LANG1:78</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:13_4_1_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><b>hence </b><a NAME="E4:13_4_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4_1_1"><i><font color="Green" title="E26">A25</font></i></a>, <a class="ref" href="zf_lang1.html#T81" title="ZF_LANG1:th.81">ZF_LANG1:81</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_1_2"/><i><font color="Green" title="E26">A26</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font></span><br/><a NAME="E1:13_4_1_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E2:13_4_1"><i><font color="Green" title="E24">A23</font></i></a>, <a class="txt" href="zfmodel2.html#E1:13_4_1_2"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><a NAME="E2:13_4_1_2_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="txt" href="zfmodel2.html#E7:13_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="txt" href="zfmodel2.html#E1:13_4_1_2"><i><font color="Green" title="E26">A26</font></i></a>, <a class="ref" href="zf_lang1.html#T79" title="ZF_LANG1:th.79">ZF_LANG1:79</a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_4_1_2_1"/>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:13_4_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:13_4_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E10:13_4"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="zfmodel2.html#E3:13_4_1"><i><font color="Green" title="E25">A24</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E12:13_4"/><i><font color="Green" title="E24">A27</font></i>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/>

<div><i><font color="Green" title="E25">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_2"/><i><font color="Green" title="E26">A29</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/><a NAME="E2:13_4_2"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E12:13_4"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="zf_lang1.html#T81" title="ZF_LANG1:th.81">ZF_LANG1:81</a></i>;<br/><a NAME="E3:13_4_2"/><b>then </b><i><font color="Green" title="E27">A30</font></i>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E4:13_4"><i><font color="Green" title="E20">A19</font></i></a>, <a class="txt" href="zfmodel2.html#E1:13_4_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="zfmodel1.html#T10" title="ZFMODEL1:th.10">ZFMODEL1:10</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/><a NAME="E1:13_4_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E3:13_4_2"><i><font color="Green" title="E27">A30</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><a NAME="E2:13_4_2_1"/><b>then </b>
( <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> )
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:13_4_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="zfmodel2.html#E5:13_4"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E5:13_4_2"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><b>hence </b><a NAME="E6:13_4_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4_2"><i><font color="Green" title="E26">A29</font></i></a>, <a class="ref" href="zf_lang1.html#T174" title="ZF_LANG1:th.174">ZF_LANG1:174</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_4_3"/><i><font color="Green" title="E26">A31</font></i>: 
( <font color="Maroon" title="c7">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">z</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span><br/><a NAME="E1:13_4_3_1"/>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E12:13_4"><i><font color="Green" title="E24">A27</font></i></a>, <a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><a NAME="E2:13_4_3_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4_3"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="zf_lang1.html#T79" title="ZF_LANG1:th.79">ZF_LANG1:79</a></i>;<br/><a NAME="E3:13_4_3_1"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E7:13_4"><i><font color="Green" title="E22">A21</font></i></a>, <a class="ref" href="zf_lang1.html#D1" title="ZF_LANG1:def.1">ZF_LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E4:13_4_3_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4"><i><font color="Green" title="E18">A17</font></i></a>, <a class="txt" href="zfmodel2.html#E5:13_4"><i><font color="Green" title="E21">A20</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E3:13_4_3"/><b>then </b>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a>  <a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="zf_lang1.html#T80" title="ZF_LANG1:th.80">ZF_LANG1:80</a></i>;<br/><b>hence </b><a NAME="E4:13_4_3"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E1:13_4_3"><i><font color="Green" title="E26">A31</font></i></a>, <a class="ref" href="zf_lang1.html#T173" title="ZF_LANG1:th.173">ZF_LANG1:173</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:13_4"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K8" title="ZF_LANG:func.8">All</a> <font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font></span>)</span> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font>
 <b>by </b><i><a class="txt" href="zfmodel2.html#E10:13_4"><i><font color="Green" title="E23">A22</font></i></a>, <a class="txt" href="zfmodel2.html#E13:13_4"><i><font color="Green" title="E25">A28</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:13"/>
for <font color="Olive" title="b1">H</font> being  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>]
 
<b>from </b><i><a class="ref" href="zf_lang1.html#S1" title="ZF_LANG1:sch.1">ZF_LANG1:sch 1</a>(<a class="txt" href="zfmodel2.html#E1:13"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="zfmodel2.html#E2:13"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="zfmodel2.html#E3:13"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="zfmodel2.html#E4:13"><i><font color="Green" title="E17">A16</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:13"/>
( not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="c4">H</font> implies ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c1">x</font> iff <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">v</font> <a href="zf_lang1.html#K1" title="ZF_LANG1:func.1">/</a> <font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c4">H</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
