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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">e</font> being  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being  <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font> holds  <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="Olive" title="b2">f1</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Olive" title="b3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Olive" title="b3">f2</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">e</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"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="abcmiz_1.html#NM13" title="ABCMIZ_1:NM.13">valuation</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Olive" title="b2">f2</font></span>)</span></span><br/>



<b>set </b><font color="Maroon" title="c3">t</font> = <font color="Maroon" title="c2">e</font>;<br/>
<b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be   <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"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">e</font> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span><br/>

<a NAME="E1:276"/><i><font color="Green" title="E128">A0</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="abcmiz_1.html#D66" target="_self" title="ABCMIZ_1:def.66">SUBST2</a></i>;<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><span class="p1">(<span class="default">$1 <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>;<br/>
<a NAME="E2:276"/><i><font color="Green" title="E129">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="276_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="c6">x</font> <a href="abcmiz_1.html#K35" title="ABCMIZ_1:func.35">-term</a> <font color="Maroon" title="c1">C</font>]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="276_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:276_1_1"/>
( <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> or <font color="Maroon" title="c6">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="276_1_1_1"><b>suppose </b></a><a NAME="E1:276_1_1_1"/><i><font color="Green" title="E130">Z0</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</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">S<sub>1</sub></font>[<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><br/><div class="add"><a NAME="E2:276_1_1_1"/><b>then </b>
<span class="p1">(<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> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><b>hence </b><a NAME="E3:276_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<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>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_1_1_1"><i><font color="Green" title="E130">Z0</font></i></a>, <a class="ref" href="abcmiz_1.html#D66" target="_self" title="ABCMIZ_1:def.66">SUBST2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="276_1_1_2"><b>suppose </b></a><a NAME="E1:276_1_1_2"/><i><font color="Green" title="E130">Z0</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</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">S<sub>1</sub></font>[<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><br/><div class="add"><a NAME="E2:276_1_1_2"/><b>then </b>
( <font color="Maroon" title="c6">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c6">x</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f2</font> )
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276"><i><font color="Green" title="E128">A0</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:276_1_1_2"/><b>then </b>
( <span class="p1">(<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> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c4">f1</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; <span class="p1">(<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> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c5">f2</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> )
 <b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>;<br/><b>hence </b><a NAME="E4:276_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<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>]
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_1_1_2"><i><font color="Green" title="E130">Z0</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></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:276"/><i><font color="Green" title="E130">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="276_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="c6">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="c6">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="c7">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="c7">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="c6">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="c7">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="c6">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c7">p</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:276_2"/><i><font color="Green" title="E131">Z0</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">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="c6">c</font></span>)</span>
 <b>and </b><br/><a NAME="E2:276_2"/><i><font color="Green" title="E132">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="c7">p</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">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="c6">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c7">p</font>]</span><br/>

<a NAME="E3:276_2"/><i><font color="Green" title="E133">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</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="c7">p</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></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="c7">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</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="E4:276_2"/><b>then </b><i><font color="Green" title="E134">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</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="c7">p</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></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="c7">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</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="c7">p</font> )
 
<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="276_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">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="c8">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="c7">p</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/><b>assume </b><a NAME="E1:276_2_1"/>
<font color="Maroon" title="c8">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="c7">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/><a NAME="E2:276_2_1"/><b>then </b><i><font color="Green" title="E135">Z3</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> &amp; <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c7">p</font> )
 <b>by </b><i><a class="txt" href="abcmiz_1.html#E4:276_2"><i><font color="Green" title="E134">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="E3:276_2_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">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="c9">pi</font> = <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="abcmiz_1.html#E2:276_2_1"><i><font color="Green" title="E135">Z3</font></i></a>, <a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th42</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">pi</font> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <font color="Maroon" title="c5">f2</font>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:276_2_1"><i><font color="Green" title="E135">Z3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">pi</font> <a href="abcmiz_1.html#K59" title="ABCMIZ_1:func.59">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:276_2"><i><font color="Green" title="E132">Z1</font></i></a>, <a class="txt" href="abcmiz_1.html#E2:276_2_1"><i><font color="Green" title="E135">Z3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E2:276_2_1"><i><font color="Green" title="E135">Z3</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:276_2"/><b>then </b><i><font color="Green" title="E135">Z4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_1.html#E4:276_2"><i><font color="Green" title="E134">Z2</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</font>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_2"><i><font color="Green" title="E131">Z0</font></i></a>, <a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
<br/>.= 
<font color="Maroon" title="c6">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="c7">p</font> <a href="abcmiz_1.html#K58" title="ABCMIZ_1:func.58">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_2"><i><font color="Green" title="E131">Z0</font></i></a>, <a class="txt" href="abcmiz_1.html#E3:276_2"><i><font color="Green" title="E133">Z5</font></i></a>, <a class="txt" href="abcmiz_1.html#E6:276_2"><i><font color="Green" title="E135">Z4</font></i></a>, <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"><font color="Maroon" title="c6">c</font> <a href="abcmiz_1.html#K36" title="ABCMIZ_1:func.36">-trm</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_2"><i><font color="Green" title="E131">Z0</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="E4:276"/><i><font color="Green" title="E131">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="276_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="c6">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="c6">a</font>] )</span><br/>

<b>assume </b><a NAME="E1:276_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">a</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>[<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="c6">a</font>]</span><br/>

<b>then </b><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="c6">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>
<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"><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="c6">a</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
;<br/>
<b>hence </b><a NAME="E3:276_3"/>
<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="c6">a</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="E5:276"/><i><font color="Green" title="E132">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="276_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="c6">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="c6">a</font>,<font color="Olive" title="b1">t</font>] )</span><br/>

<b>assume </b><a NAME="E1:276_4"/><i><font color="Green" title="E133">Z1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">a</font>,<font color="Olive" title="b1">t</font>]</span><br/>

<b>let </b><font color="Maroon" title="c7">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="c7">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="c6">a</font>,<font color="Maroon" title="c7">t</font>] )</span><br/>

<b>assume </b><a NAME="E2:276_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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>[<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="c6">a</font>,<font color="Maroon" title="c7">t</font>]</span><br/>

<b>then </b><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="c6">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <font color="Maroon" title="c4">f1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="abcmiz_1.html#K62" title="ABCMIZ_1:func.62">at</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <font color="Maroon" title="c5">f2</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K60" title="ABCMIZ_1:func.60">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">t</font> <a href="abcmiz_1.html#K62" title="ABCMIZ_1:func.62">at</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_1.html#E1:276_4"><i><font color="Green" title="E133">Z1</font></i></a>, <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"><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="c6">a</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="abcmiz_1.html#K56" title="ABCMIZ_1:func.56">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c5">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="abcmiz_1.html#D60" target="_self" title="ABCMIZ_1:def.60">SUBST</a></i>
;<br/>
<b>hence </b><a NAME="E4:276_4"/>
<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="c6">a</font>,<font color="Maroon" title="c7">t</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>
<b>thus </b><a NAME="E6:276"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">e</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="abcmiz_1.html#E2:276"><i><font color="Green" title="E129">W1</font></i></a>, <a class="txt" href="abcmiz_1.html#E3:276"><i><font color="Green" title="E130">W2</font></i></a>, <a class="txt" href="abcmiz_1.html#E4:276"><i><font color="Green" title="E131">W3</font></i></a>, <a class="txt" href="abcmiz_1.html#E5:276"><i><font color="Green" title="E132">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>
