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<div class="add">

<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="abcmiz_1.html#V3" title="ABCMIZ_1:attr.3">initialized</a> <a href="abcmiz_1.html#NM5" title="ABCMIZ_1:NM.5">ConstructorSignature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> <a href="abcmiz_1.html#V15" title="ABCMIZ_1:attr.15">irrelevant</a> <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font>  st ( for <font color="Olive" title="b3">x</font>, <font color="Olive" title="b4">y</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Olive" title="b4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> ) ) holds <br/>for <font color="Olive" title="b3">e</font> being  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>  st  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b3">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e</font></span><br/><b>let </b><font color="Maroon" title="c2">f</font>, <font color="Maroon" title="c3">g</font> be  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> <a href="abcmiz_1.html#V15" title="ABCMIZ_1:attr.15">irrelevant</a> <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> ) ) implies for <font color="Olive" title="b1">e</font> being  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>  st  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> )</span><br/>



<b>assume </b><a NAME="E1:264"/><i><font color="Green" title="E123">A0</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>  st  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font></span><br/>

<b>let </b><font color="Maroon" title="c4">t</font> be   <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c4">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>] <b>means </b>(  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font> implies <span class="p1">(<span class="default">$1 <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 );<br/>
<a NAME="E2:264"/><i><font color="Green" title="E124">W1</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> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="264_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>]</span><br/>

<b>assume </b><a NAME="E1:264_1"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span><br/>

<a NAME="E2:264_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="msafree3.html#T11" title="MSAFREE3:th.11">MSAFREE3:11</a></i>;<br/>
<a NAME="E3:264_1"/><b>then </b><i><font color="Green" title="E125">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">y</font> being   <a href="abcmiz_1.html#NM3" title="ABCMIZ_1:NM.3">variable</a><b> such that </b><br/><a NAME="E5:264_1"/><i><font color="Green" title="E126">A3</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D58" target="_self" title="ABCMIZ_1:def.58">IRR</a></i>;<br/>
<a NAME="E6:264_1"/>
( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">g</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font> )
 
<b>by </b><i><a class="txt" href="#E1:264"><i><font color="Green" title="E123">A0</font></i></a>, <a class="txt" href="#E3:264_1"><i><font color="Green" title="E125">A2</font></i></a>, <a class="txt" href="#E5:264_1"><i><font color="Green" title="E126">A3</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/>
<b>hence </b><a NAME="E7:264_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:264"/><i><font color="Green" title="E125">W2</font></i>: 
for <font color="Olive" title="b1">c</font> being <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Olive" title="b1">c</font></span>)</span> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="264_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">c</font> be  <a href="abcmiz_1.html#V2" title="ABCMIZ_1:attr.2">constructor</a> <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">c</font></span>)</span> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c6">p</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">c</font></span>)</span> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:264_2"/><i><font color="Green" title="E126">Z0</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c5">c</font></span>)</span>
 <b>and </b><br/><a NAME="E2:264_2"/><i><font color="Green" title="E127">Z1</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 <b>and </b><br/><a NAME="E3:264_2"/><i><font color="Green" title="E128">00</font></i>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E4:264_2"/>
<font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c6">p</font>
 
<b>by </b><i><a class="txt" href="#E1:264_2"><i><font color="Green" title="E126">Z0</font></i></a>, <a class="ref" href="abcmiz_1.html#D35" target="_self" title="ABCMIZ_1:def.35">TERM</a></i>;<br/>
<a NAME="E5:264_2"/><b>then </b><i><font color="Green" title="E129">01</font></i>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T88" target="_self" title="ABCMIZ_1:th.88">Th84</a></i>;<br/>
<a NAME="E6:264_2"/><i><font color="Green" title="E130">Z5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T130" target="_self" title="ABCMIZ_1:th.130">ThTr2</a></i>;<br/>
<a NAME="E7:264_2"/><b>then </b><i><font color="Green" title="E131">Z2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="264_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> )</span><br/><b>assume </b><a NAME="E1:264_2_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span><br/><a NAME="E2:264_2_1"/><b>then </b><i><font color="Green" title="E132">Z3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="abcmiz_1.html#K55" title="ABCMIZ_1:func.55">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">g</font> <a href="abcmiz_1.html#K55" title="ABCMIZ_1:func.55">#</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> &amp; <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> )
 <b>by </b><i><a class="txt" href="#E7:264_2"><i><font color="Green" title="E131">Z2</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:264_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c8">pi</font> = <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> as   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> <b>by </b><i><a class="txt" href="#E2:264_2_1"><i><font color="Green" title="E132">Z3</font></i></a>, <a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/><a NAME="E5:264_2_1"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c8">pi</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">s</font></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">p</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="#E2:264_2_1"><i><font color="Green" title="E132">Z3</font></i></a></i>;<br/><a NAME="E6:264_2_1"/><b>then </b><i><font color="Green" title="E133">Z4</font></i>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c8">pi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:264_2"><i><font color="Green" title="E129">01</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:264_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">pi</font> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c3">g</font>
 <b>by </b><i><a class="txt" href="#E2:264_2_1"><i><font color="Green" title="E132">Z3</font></i></a></i>;<br/><b>hence </b><a NAME="E8:264_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
 <b>by </b><i><a class="txt" href="#E2:264_2"><i><font color="Green" title="E127">Z1</font></i></a>, <a class="txt" href="#E2:264_2_1"><i><font color="Green" title="E132">Z3</font></i></a>, <a class="txt" href="#E6:264_2_1"><i><font color="Green" title="E133">Z4</font></i></a>, <a class="txt" href="#E3:264_2"><i><font color="Green" title="E128">00</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E9:264_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="#E1:264_2"><i><font color="Green" title="E126">Z0</font></i></a>, <a class="txt" href="#E7:264_2"><i><font color="Green" title="E131">Z2</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>hence </b><a NAME="E10:264_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="#E1:264_2"><i><font color="Green" title="E126">Z0</font></i></a>, <a class="txt" href="#E6:264_2"><i><font color="Green" title="E130">Z5</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:264"/><i><font color="Green" title="E126">W3</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Olive" title="b1">a</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="264_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</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="c5">a</font>] implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:264_3"/><i><font color="Green" title="E127">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>]
 <b>and </b><br/><a NAME="E2:264_3"/><i><font color="Green" title="E128">A5</font></i>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font></span><br/>

<a NAME="E3:264_3"/><i><font color="Green" title="E129">A6</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="abcmiz_1.html#K10" title="ABCMIZ_1:func.10">non_op</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T43" target="_self" title="ABCMIZ_1:th.43">ThNon</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font>
<b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K32" title="ABCMIZ_1:func.32">non_op</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K30" title="ABCMIZ_1:func.30">term</a> <font color="Maroon" title="c5">a</font>
<b>by </b><i><a class="txt" href="#E1:264_3"><i><font color="Green" title="E127">A4</font></i></a>, <a class="txt" href="#E2:264_3"><i><font color="Green" title="E128">A5</font></i></a>, <a class="txt" href="#E3:264_3"><i><font color="Green" title="E129">A6</font></i></a>, <a class="ref" href="abcmiz_1.html#T93" target="_self" title="ABCMIZ_1:th.93">Aux1</a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:264"/><i><font color="Green" title="E127">W4</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>] holds <br/>for <font color="Olive" title="b2">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="264_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</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="c5">a</font>] implies for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">t</font>] )</span><br/>

<b>assume </b><a NAME="E1:264_4"/><i><font color="Green" title="E128">A7</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">a</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Olive" title="b1">t</font>]</span><br/>

<b>let </b><font color="Maroon" title="c6">t</font> be    <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">t</font>] implies <font color="Maroon">S<sub>1</sub></font>[<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font>] )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:264_4"/><i><font color="Green" title="E129">A8</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">t</font>]
 <b>and </b><br/><a NAME="E3:264_4"/><i><font color="Green" title="E130">A9</font></i>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span><br/>

<a NAME="E4:264_4"/>
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="abcmiz_1.html#K9" title="ABCMIZ_1:func.9">*</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T46" target="_self" title="ABCMIZ_1:th.46">ThAst</a></i>;<br/>
<b>then </b><i><font color="Green" title="E131">AA</font></i>:  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="pboole.html#K2" title="PBOOLE:func.2">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c6">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="abcmiz_1.html#K8" title="ABCMIZ_1:func.8">a_Term</a> 
<b>by </b><i><a class="ref" href="abcmiz_1.html#T96" target="_self" title="ABCMIZ_1:th.96">Aux2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c6">t</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D7" title="PBOOLE:def.7">PBOOLE:def 7</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c2">f</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="abcmiz_1.html#K62" title="ABCMIZ_1:func.62">at</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c3">g</font>
<b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K33" title="ABCMIZ_1:func.33">ast</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="abcmiz_1.html#K31" title="ABCMIZ_1:func.31">term</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">t</font>
<b>by </b><i><a class="txt" href="#E1:264_4"><i><font color="Green" title="E128">A7</font></i></a>, <a class="txt" href="#E2:264_4"><i><font color="Green" title="E129">A8</font></i></a>, <a class="txt" href="#E5:264_4"><i><font color="Green" title="E131">AA</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a>, <a class="txt" href="#E3:264_4"><i><font color="Green" title="E130">A9</font></i></a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</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>thus </b><a NAME="E6:264"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">t</font>]
 <b>from </b><i><a class="ref" href="abcmiz_1.html#S2" target="_self" title="ABCMIZ_1:sch.2">ABCMIZ_1:sch 2</a>(<a class="txt" href="#E2:264"><i><font color="Green" title="E124">W1</font></i></a>, <a class="txt" href="#E3:264"><i><font color="Green" title="E125">W2</font></i></a>, <a class="txt" href="#E4:264"><i><font color="Green" title="E126">W3</font></i></a>, <a class="txt" href="#E5:264"><i><font color="Green" title="E127">W4</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
