<?xml version="1.0"?>
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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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">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="b3">p</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b2">e</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="Olive" title="b1">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="Olive" title="b3">p</font> holds <br/> <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b2">e</font> <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="b4">t</font></span>)</span> where <font color="Olive" title="b4">t</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="b4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">p</font> </span>}</span> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">p</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">e</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="c2">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="Olive" title="b2">p</font> holds <br/> <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">e</font> <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="b3">t</font></span>)</span> where <font color="Olive" title="b3">t</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="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">p</font> </span>}</span> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Maroon" title="c3">e</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="c2">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="Olive" title="b1">p</font> holds <br/> <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> <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="b2">t</font></span>)</span> where <font color="Olive" title="b2">t</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="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">p</font> </span>}</span> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="trees_3.html#V6" title="TREES_3:attr.6">DTree-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">e</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="c2">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="c4">p</font> implies  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> <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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span>  )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> ;<br/>
<span class="kw">assume </span><a NAME="E1:171"/><span class="lab"><font color="Green" title="E84">A1</font></span>: 
<font color="Maroon" title="c3">e</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="c2">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="c4">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> <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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> </span><br/>

<a NAME="E2:171"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K34" title="ABCMIZ_1:func.34">QuasiTerms</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T51" target="_self" title="ABCMIZ_1:th.51">Th51</a></span>;<br/>
<a NAME="E3:171"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> is    <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E4:171"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A2</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:171"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span>  <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> <font color="Maroon" title="c3">e</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="171_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:171_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><span class="kw"> such that </span><br/><a NAME="E3:171_1"/><span class="lab"><font color="Green" title="E86">A3</font></span>: 
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font>
 <span class="kw">and </span><br/><a NAME="E4:171_1"/><span class="lab"><font color="Green" title="E87">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<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="c7">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K14" title="ABCMIZ_1:func.14">a_Term</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="ref" href="msafree3.html#T12" title="MSAFREE3:th.12">MSAFREE3:12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c7">t</font> as   <a href="abcmiz_1.html#NM9" title="ABCMIZ_1:NM.9">quasi-term</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:171"><span class="lab"><font color="Green" title="E85">A2</font></span></a>, <a class="txt" href="#E3:171_1"><span class="lab"><font color="Green" title="E86">A3</font></span></a>, <a class="ref" href="abcmiz_1.html#T41" target="_self" title="ABCMIZ_1:th.41">Th41</a></span>;<br/>
<a NAME="E6:171_1"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c8">t</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:171_1"><span class="lab"><font color="Green" title="E86">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:171_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:171_1"><span class="lab"><font color="Green" title="E87">A4</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span>  or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:171"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E8:171"/><span class="lab"><font color="Green" title="E86">A5</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font>
 <span class="kw">and </span><br/><a NAME="E9:171"/><span class="lab"><font color="Green" title="E87">A6</font></span>: 
<font color="Maroon" title="c7">Y</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">t</font></span>)</span> where <font color="Olive" title="b1">t</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">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<a NAME="E10:171"/>
 ex <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 <br/>( <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">t</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:171"><span class="lab"><font color="Green" title="E87">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:171"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c3">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:171"><span class="lab"><font color="Green" title="E84">A1</font></span></a>, <a class="txt" href="#E8:171"><span class="lab"><font color="Green" title="E86">A5</font></span></a>, <a class="ref" href="msafree3.html#T12" title="MSAFREE3:th.12">MSAFREE3:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
