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<div class="add">

<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">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span><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">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Olive" title="b2">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Olive" title="b3">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">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/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span>; <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">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">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Olive" title="b1">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">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/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span></span><br/>





<span class="kw">thus </span><a NAME="E1:275"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</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"> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="275_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:275_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E3:275_1"/><span class="lab"><font color="Green" title="E127">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">and </span><br/><a NAME="E4:275_1"/><span class="lab"><font color="Green" title="E128">A2</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">b</font> being   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E6:275_1"/><span class="lab"><font color="Green" title="E129">A3</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c3">f1</font>
 <span class="kw">and </span><br/><a NAME="E7:275_1"/><span class="lab"><font color="Green" title="E130">A4</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:275_1"><span class="lab"><font color="Green" title="E128">A2</font></span></a></span>;<br/>
<a NAME="E8:275_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:275_1"><span class="lab"><font color="Green" title="E127">A1</font></span></a>, <a class="txt" href="#E6:275_1"><span class="lab"><font color="Green" title="E129">A3</font></span></a>, <a class="ref" href="abcmiz_1.html#T149" target="_self" title="ABCMIZ_1:th.149">Th149</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:275_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:275_1"><span class="lab"><font color="Green" title="E130">A4</font></span></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="c5">x</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:275"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font><span class="kw"> such that </span><br/><a NAME="E4:275"/><span class="lab"><font color="Green" title="E127">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="abcmiz_1.html#K65" title="ABCMIZ_1:func.65">at</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:275"/><span class="lab"><font color="Green" title="E128">A6</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>
<a NAME="E6:275"/><span class="lab"><font color="Green" title="E129">A7</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:275"><span class="lab"><font color="Green" title="E127">A5</font></span></a>, <a class="ref" href="abcmiz_1.html#T149" target="_self" title="ABCMIZ_1:th.149">Th149</a></span>;<br/>
<a NAME="E7:275"/>
<font color="Maroon" title="c6">a</font> <a href="abcmiz_1.html#K61" title="ABCMIZ_1:func.61">at</a> <font color="Maroon" title="c3">f1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:275"><span class="lab"><font color="Green" title="E128">A6</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:275"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="abcmiz_1.html#K63" title="ABCMIZ_1:func.63">at</a> <font color="Maroon" title="c4">f2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:275"><span class="lab"><font color="Green" title="E129">A7</font></span></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>
