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

<b>set </b><font color="Maroon" title="c3">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>;<br/>
<div><i><font color="Green" title="E117">AA</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="242_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">t</font>]</span><br/><b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> as    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="ref" href="abcmiz_1.html#D28" target="_self" title="ABCMIZ_1:def.28">ELEMENT</a></i>;<br/><b>take </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">t</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">t</font>]</span><br/><b>thus </b><a NAME="E2:242_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">t</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="abcmiz_1.html#K2" title="ABCMIZ_1:func.2">Vars</a> , <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E3:242_1_1"/><i><font color="Green" title="E118">A0</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3" title="FUNCT_2:sch.3">FUNCT_2:sch 3</a>(<a class="txt" href="#E1:242_1_1"><i><font color="Green" title="E117">AA</font></i></a>);<br/></i>
<a NAME="E4:242_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="242_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span>  or <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g</font> )</span><br/>

<b>assume </b><a NAME="E1:242_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g</font></span><br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a><b> such that </b><br/><a NAME="E3:242_1_1_2"/><i><font color="Green" title="E119">A2</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 ;<br/>
<a NAME="E4:242_1_1_2"/>
( <font color="Maroon" title="c4">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K2" title="ABCMIZ_1:func.2">Vars</a>  )
 
<b>by </b><i><a class="txt" href="#E3:242_1_1"><i><font color="Green" title="E118">A0</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:242_1_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">g</font>
 <b>by </b><i><a class="txt" href="#E3:242_1_1_2"><i><font color="Green" title="E119">A2</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:242_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">x</font>,<span class="p3">(<span class="default"><font color="Olive" title="b1">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> where <font color="Olive" title="b1">x</font> is   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a> : <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> </span>}</span>  is   <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
