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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">M</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="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#K2" title="ZF_LANG1:func.2">/</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/><span class="kw">let </span><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#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b2">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Olive" title="b1">H</font> holds <br/>( <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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/><span class="kw">let </span><font color="Maroon" title="c4">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_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#K2" title="ZF_LANG1:func.2">/</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/>









<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>] <span class="kw">means </span>(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> $1 implies  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#K2" title="ZF_LANG1:func.2">/</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"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Olive" title="b2">x2</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Olive" title="b2">x2</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</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#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c7">x2</font>] )</span><br/>



<a NAME="E1:13_1"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
( <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/>
<a NAME="E2:13_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
( <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> )
 
;<br/>
<span class="kw">thus </span><a NAME="E3:13_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c7">x2</font>]</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_1_1"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K3" title="ZF_LANG:func.3">'='</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#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K3" title="ZF_LANG:func.3">'='</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#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:13_1_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
( ( <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> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<a NAME="E4:13_1_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/>
<a NAME="E5:13_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/>
<a NAME="E6:13_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K3" title="ZF_LANG:func.3">'='</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c10">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13_1_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="zf_lang1.html#T152" onmouseover="rs('zf_lang1/T152')" onmouseout="rh()">ZF_LANG1:152</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:13_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#K3" title="ZF_LANG:func.3">'='</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#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</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#K3" title="ZF_LANG:func.3">'='</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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#K3" title="ZF_LANG:func.3">'='</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font></span><br/>

<a NAME="E2:13_1_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E4:13_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E8:13_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K3" title="ZF_LANG:func.3">'='</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" 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#K3" title="ZF_LANG:func.3">'='</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="E9:13_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:13_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#K3" title="ZF_LANG:func.3">'='</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E5:13_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E4:13_1_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="ref" href="zf_model.html#T12" onmouseover="rs('zf_model/T12')" onmouseout="rh()">ZF_MODEL:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">consider </span><font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><br/><a NAME="E5:13_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
( ( <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="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c9">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="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c9">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="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c9">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="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:13_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:13_1"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c7">x2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">'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#K2" title="ZF_LANG1:func.2">/</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">'in'</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<a NAME="E7:13_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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="c10">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/>
<a NAME="E8:13_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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="c10">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a>, <a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/>
<a NAME="E9:13_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'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="c8">y1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'in'</a> <font color="Maroon" title="c9">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="zf_lang1.html#T154" onmouseover="rs('zf_lang1/T154')" onmouseout="rh()">ZF_LANG1:154</a></span>;<br/>
<span class="kw">thus </span><a NAME="E10:13_1"/>
( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font></span><br/>

<a NAME="E2:13_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">v</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:13_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:13_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E11:13_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">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">'in'</a> <font color="Maroon" title="c7">x2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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="E12:13_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:13_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c10">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">'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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:13_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="txt" href="#E7:13_1"><span class="lab"><font color="Green" title="E20">A9</font></span></a>, <a class="txt" href="#E8:13_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="zf_model.html#T13" onmouseover="rs('zf_model/T13')" onmouseout="rh()">ZF_MODEL:13</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:13"/><span class="lab"><font color="Green" title="E17">A12</font></span>: 
 for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">H2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Olive" title="b2">H2</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_2"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_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#K2" title="ZF_LANG1:func.2">/</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> ) )
 <span class="kw">and </span><br/><a NAME="E2:13_2"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font>]</span><br/>

<span class="kw">assume </span><a NAME="E3:13_2"/>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K6" title="ZF_LANG:func.6">'&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#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )</span><br/>

<a NAME="E4:13_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A15</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T141" onmouseover="rs('zf_lang1/T141')" onmouseout="rh()">ZF_LANG1:141</a></span>;<br/>
<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K6" title="ZF_LANG:func.6">'&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#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:13_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">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&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#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&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#K6" title="ZF_LANG:func.6">'&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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_2_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#K6" title="ZF_LANG:func.6">'&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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font></span><br/>

<a NAME="E2:13_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<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#K6" title="ZF_LANG:func.6">'&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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T158" onmouseover="rs('zf_lang1/T158')" onmouseout="rh()">ZF_LANG1:158</a></span>;<br/>
<a NAME="E3:13_2_1"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E4:13_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A17</font></span>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E4:13_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:13_2_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> <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_2_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E6:13_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E4:13_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:13_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_2_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E6:13_2"/><span class="lab"><font color="Green" title="E21">A18</font></span>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K6" title="ZF_LANG:func.6">'&amp;'</a> <font color="Maroon" title="c7">H2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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#K6" title="ZF_LANG:func.6">'&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_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E8:13_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A19</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_2"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="#E4:13_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:13_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c8">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_2"><span class="lab"><font color="Green" title="E21">A18</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<a NAME="E10:13_2"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_2"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E4:13_2"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E11:13_2"/><span class="kw">then </span>
<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#K6" title="ZF_LANG:func.6">'&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_2"><span class="lab"><font color="Green" title="E22">A19</font></span></a>, <a class="ref" href="zf_model.html#T15" onmouseover="rs('zf_model/T15')" onmouseout="rh()">ZF_MODEL:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:13_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">H1</font> <a href="zf_lang.html#K6" title="ZF_LANG:func.6">'&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 class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T158" onmouseover="rs('zf_lang1/T158')" onmouseout="rh()">ZF_LANG1:158</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:13"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
 for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a><br/>  for <font color="Olive" title="b2">z</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b2">z</font>,<font color="Olive" title="b1">H</font>)]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c6">H</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">z</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:13_3"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)]</span><br/>

<a NAME="E2:13_3"/>
( <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/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">s</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><span class="kw"> such that </span><br/><a NAME="E4:13_3"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
( ( <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/>
<span class="kw">assume </span><a NAME="E5:13_3"/><span class="lab"><font color="Green" title="E21">A23</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K7" title="ZF_LANG:func.7">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#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>) )</span><br/>

<a NAME="E6:13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A24</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T142" onmouseover="rs('zf_lang1/T142')" onmouseout="rh()">ZF_LANG1:142</a></span>;<br/>
<a NAME="E7:13_3"/><span class="kw">then </span>
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E8:13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A25</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K7" title="ZF_LANG:func.7">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#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>) )</span><br/>

<a NAME="E9:13_3"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T151" onmouseover="rs('zf_lang1/T151')" onmouseout="rh()">ZF_LANG1:151</a></span>;<br/>
<a NAME="E10:13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A26</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <font color="Maroon" title="c6">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_3"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:13_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#K7" title="ZF_LANG:func.7">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#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">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#K7" title="ZF_LANG:func.7">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_3_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#K7" title="ZF_LANG:func.7">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span><br/>

<a NAME="E2:13_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A27</font></span>: 
<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#K7" title="ZF_LANG:func.7">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_3"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="ref" href="zf_lang1.html#T159" onmouseover="rs('zf_lang1/T159')" onmouseout="rh()">ZF_LANG1:159</a>, <a class="ref" href="zf_lang1.html#T160" onmouseover="rs('zf_lang1/T160')" onmouseout="rh()">ZF_LANG1:160</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E26">A28</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>) )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_3_1_1"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E2:13_3_1_1"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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_3_1_1_1"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/><a NAME="E2:13_3_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_3_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:13_3_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E3:13_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E6:13_3"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E4:13_3_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_3"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E1:13_3_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="funct_7.html#T33" onmouseover="rs('funct_7/T33')" onmouseout="rh()">FUNCT_7:33</a></span>;<br/><a NAME="E5:13_3_1_1_1"/><span class="kw">then </span>
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:13_3"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="txt" href="#E3:13_3_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E1:13_3_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="ref" href="zfmodel1.html#T10" onmouseover="rs('zfmodel1/T10')" onmouseout="rh()">ZFMODEL1:10</a></span>;<br/><a NAME="E6:13_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E7:13_3_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T34" onmouseover="rs('funct_7/T34')" onmouseout="rh()">FUNCT_7:34</a></span>;<br/><span class="kw">hence </span><a NAME="E8:13_3_1_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13_3_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="funct_7.html#T35" onmouseover="rs('funct_7/T35')" onmouseout="rh()">FUNCT_7:35</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:13_3_1_1"/><span class="kw">then </span>
<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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><span class="kw">hence </span><a NAME="E5:13_3_1_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="ref" href="zf_lang1.html#T72" onmouseover="rs('zf_lang1/T72')" onmouseout="rh()">ZF_LANG1:72</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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> implies <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>) )</span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_3_1_2"/><span class="lab"><font color="Green" title="E27">A34</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:13_3_1_2"/><span class="lab"><font color="Green" title="E28">A35</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds  <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#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c7">z</font>,<font color="Olive" title="b1">m</font>) <a href="zf_model.html#R1" title="ZF_MODEL:pred.1">|=</a> <font color="Maroon" title="c6">H</font></span></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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_3_1_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13_3_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:13_3_1_2"><span class="lab"><font color="Green" title="E28">A35</font></span></a>, <a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E2:13_3_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A36</font></span>: 
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E6:13_3"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:13_3_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_3"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>;<br/><span class="kw">hence </span><a NAME="E4:13_3_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#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3_1_2"><span class="lab"><font color="Green" title="E27">A34</font></span></a>, <a class="txt" href="#E2:13_3_1_2_1"><span class="lab"><font color="Green" title="E29">A36</font></span></a>, <a class="ref" href="funct_7.html#T33" onmouseover="rs('funct_7/T33')" onmouseout="rh()">FUNCT_7:33</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:13_3_1_2"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:13_3_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_3"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E3:13_3_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E12:13_3"/><span class="lab"><font color="Green" title="E25">A37</font></span>: 
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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#K7" title="ZF_LANG:func.7">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="E13:13_3"/>
 <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T151" onmouseover="rs('zf_lang1/T151')" onmouseout="rh()">ZF_LANG1:151</a></span>;<br/>
<a NAME="E14:13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A38</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_model.html#K2" title="ZF_MODEL:func.2">Free</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c6">H</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13_3"><span class="lab"><font color="Green" title="E21">A23</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E27">A39</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">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> 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#K7" title="ZF_LANG:func.7">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></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_3_2"/><span class="lab"><font color="Green" title="E28">A40</font></span>: 
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E2:13_3_2"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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#K7" title="ZF_LANG:func.7">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="E3:13_3_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>  <a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:13_3"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="txt" href="#E1:13_3_2"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="ref" href="zf_lang1.html#T72" onmouseover="rs('zf_lang1/T72')" onmouseout="rh()">ZF_LANG1:72</a></span>;<br/><a NAME="E4:13_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A41</font></span>: 
<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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c1">x</font>,<span class="p1">(<span class="default"><a href="zf_lang.html#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13_3"><span class="lab"><font color="Green" title="E26">A38</font></span></a>, <a class="txt" href="#E1:13_3_2"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="ref" href="zfmodel1.html#T10" onmouseover="rs('zfmodel1/T10')" onmouseout="rh()">ZFMODEL1:10</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">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></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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_3_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K7" title="ZF_LANG:func.7">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c6">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_3_2"><span class="lab"><font color="Green" title="E29">A41</font></span></a>, <a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E2:13_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A42</font></span>: 
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E3:13_3_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T128" onmouseover="rs('funct_7/T128')" onmouseout="rh()">FUNCT_7:128</a></span>;<br/><span class="kw">hence </span><a NAME="E4:13_3_2_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E6:13_3"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E2:13_3_2_1"><span class="lab"><font color="Green" title="E30">A42</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E6:13_3_2"/><span class="kw">then </span>
<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#K7" title="ZF_LANG:func.7">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><span class="kw">hence </span><a NAME="E7:13_3_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#K7" title="ZF_LANG:func.7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3_2"><span class="lab"><font color="Green" title="E28">A40</font></span></a>, <a class="ref" href="zf_lang1.html#T160" onmouseover="rs('zf_lang1/T160')" onmouseout="rh()">ZF_LANG1:160</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> 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#K7" title="ZF_LANG:func.7">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></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:13_3_3"/><span class="lab"><font color="Green" title="E28">A43</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:13_3_3"/>
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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#K7" title="ZF_LANG:func.7">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_3_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">M</font> holds  <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c7">z</font>,<font color="Olive" title="b1">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></a><div class="add"><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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_3_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#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E12:13_3"><span class="lab"><font color="Green" title="E25">A37</font></span></a>, <a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><a NAME="E2:13_3_3_1"/><span class="kw">then </span>
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3_3"><span class="lab"><font color="Green" title="E28">A43</font></span></a>, <a class="ref" href="funct_7.html#T33" onmouseover="rs('funct_7/T33')" onmouseout="rh()">FUNCT_7:33</a></span>;<br/><a NAME="E3:13_3_3_1"/><span class="kw">then </span>
<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#K2" title="ZF_LANG1:func.2">/</a> (<font color="Maroon" title="c7">z</font>,<font color="Maroon" title="c10">m</font>)</span>)</span> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K2" title="ZF_LANG1:func.2">/</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:13_3"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="funct_7.html#T32" onmouseover="rs('funct_7/T32')" onmouseout="rh()">FUNCT_7:32</a></span>;<br/><span class="kw">hence </span><a NAME="E4:13_3_3_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c9">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E6:13_3"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E4:13_3_3"/><span class="kw">then </span>
<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#K7" title="ZF_LANG:func.7">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T71" onmouseover="rs('zf_lang1/T71')" onmouseout="rh()">ZF_LANG1:71</a></span>;<br/><span class="kw">hence </span><a NAME="E5:13_3_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#K7" title="ZF_LANG:func.7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_3_3"><span class="lab"><font color="Green" title="E28">A43</font></span></a>, <a class="ref" href="zf_lang1.html#T159" onmouseover="rs('zf_lang1/T159')" onmouseout="rh()">ZF_LANG1:159</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E17:13_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#K7" title="ZF_LANG:func.7">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13_3"><span class="lab"><font color="Green" title="E20">A22</font></span></a>, <a class="txt" href="#E15:13_3"><span class="lab"><font color="Green" title="E27">A39</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:13"/><span class="lab"><font color="Green" title="E19">A44</font></span>: 
 for <font color="Olive" title="b1">H</font> being   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">H</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Olive" title="b1">H</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">H</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:13_4"/><span class="lab"><font color="Green" title="E20">A45</font></span>: 
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:13_4"/><span class="lab"><font color="Green" title="E21">A46</font></span>: 
 not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <span class="p1">(<span class="default"><a href="zf_lang.html#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K5" title="ZF_LANG:func.5">'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#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font> )</span><br/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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#K5" title="ZF_LANG:func.5">'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#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:13_4"/>
( <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#K5" title="ZF_LANG:func.5">'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#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font> )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'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#K5" title="ZF_LANG:func.5">'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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="13_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:13_4_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#K5" title="ZF_LANG:func.5">'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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font></span><br/>

<a NAME="E2:13_4_1"/><span class="kw">then </span>
<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#K5" title="ZF_LANG:func.5">'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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T156" onmouseover="rs('zf_lang1/T156')" onmouseout="rh()">ZF_LANG1:156</a></span>;<br/>
<a NAME="E3:13_4_1"/><span class="kw">then </span>
 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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/>
<a NAME="E4:13_4_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_4"><span class="lab"><font color="Green" title="E20">A45</font></span></a>, <a class="txt" href="#E2:13_4"><span class="lab"><font color="Green" title="E21">A46</font></span></a>, <a class="ref" href="zf_lang1.html#T140" onmouseover="rs('zf_lang1/T140')" onmouseout="rh()">ZF_LANG1:140</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:13_4_1"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E4:13_4"/>
<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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#K5" title="ZF_LANG:func.5">'not'</a> <font color="Maroon" title="c6">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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#K5" title="ZF_LANG:func.5">'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_4"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c7">v</font> <a href="zf_lang1.html#K2" title="ZF_LANG1:func.2">/</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/>
<a NAME="E6:13_4"/><span class="kw">then </span>
 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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13_4"><span class="lab"><font color="Green" title="E20">A45</font></span></a>, <a class="txt" href="#E2:13_4"><span class="lab"><font color="Green" title="E21">A46</font></span></a>, <a class="ref" href="zf_lang1.html#T140" onmouseover="rs('zf_lang1/T140')" onmouseout="rh()">ZF_LANG1:140</a></span>;<br/>
<a NAME="E7:13_4"/><span class="kw">then </span>
<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#K5" title="ZF_LANG:func.5">'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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_model.html#T14" onmouseover="rs('zf_model/T14')" onmouseout="rh()">ZF_MODEL:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:13_4"/>
<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#K5" title="ZF_LANG:func.5">'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 class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T156" onmouseover="rs('zf_lang1/T156')" onmouseout="rh()">ZF_LANG1:156</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:13"/>
 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>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="zf_lang1.html#S1" onmouseover="rs('zf_lang1/S1')" onmouseout="rh()">ZF_LANG1:sch 1</a>(<a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E19">A44</font></span></a>, <a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E17">A12</font></span></a>, <a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E18">A20</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:13"/>
(  not <font color="Maroon" title="c1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="zf_lang1.html#K4" title="ZF_LANG1:func.4">variables_in</a> <font color="Maroon" title="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#K2" title="ZF_LANG1:func.2">/</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
