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<span class="kw">let </span><font color="Maroon" title="c1">a1</font>, <font color="Maroon" title="c2">a2</font>, <font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">A</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp;  not <font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp;  not <font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">A</font> &amp; <font color="Maroon" title="c1">a1</font> is    <a href="nomin_1.html#M2" title="NOMIN_1:mode.2">TypeSCNominativeData</a> of <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">A</font> holds <br/> <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c1">a1</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c6">V</font>, <font color="Maroon" title="c7">A</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font> &amp; <font color="Maroon" title="c2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp;  not <font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp;  not <font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c1">a1</font> is    <a href="nomin_1.html#M2" title="NOMIN_1:mode.2">TypeSCNominativeData</a> of <font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font> implies  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c1">a1</font>) )</span><br/>













<span class="kw">set </span><font color="Maroon" title="c8">d1</font> = <font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">d2</font> = <font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:100"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">and </span><br/><a NAME="E2:100"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">v1</font>
 <span class="kw">and </span><br/><a NAME="E3:100"/><span class="lab"><font color="Green" title="E57">A3</font></span>: 
<font color="Maroon" title="c2">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E4:100"/><span class="lab"><font color="Green" title="E58">A4</font></span>: 
 not <font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E5:100"/><span class="lab"><font color="Green" title="E59">A5</font></span>: 
 not <font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <span class="kw">and </span><br/><a NAME="E6:100"/><span class="lab"><font color="Green" title="E60">A6</font></span>: 
<font color="Maroon" title="c1">a1</font> is    <a href="nomin_1.html#M2" title="NOMIN_1:mode.2">TypeSCNominativeData</a> of <font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c1">a1</font>)</span><br/>

<a NAME="E7:100"/><span class="lab"><font color="Green" title="E61">A7</font></span>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E8:100"/>
<font color="Maroon" title="c5">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E9:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A8</font></span>: 
 <a href="nomin_1.html#K4" title="NOMIN_1:func.4">ND_ex_1</a> (<font color="Maroon" title="c5">v2</font>,<font color="Maroon" title="c2">a2</font>) is    <a href="nomin_1.html#M2" title="NOMIN_1:mode.2">TypeSCNominativeData</a> of <font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E3:100"><span class="lab"><font color="Green" title="E57">A3</font></span></a>, <a class="ref" href="nomin_1.html#T46" target="_self" onmouseover="rs('nomin_1/T46')" onmouseout="rh()">Th46</a></span>;<br/>
<a NAME="E10:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A9</font></span>: 
 <a href="nomin_1.html#K13" title="NOMIN_1:func.13">naming</a> (<font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c3">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="txt" href="#E7:100"><span class="lab"><font color="Green" title="E61">A7</font></span></a>, <a class="ref" href="nomin_1.html#D13" target="_self" onmouseover="rs('nomin_1/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E11:100"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="100_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:100_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font></span><br/>

<a NAME="E2:100_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v1</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/>
<a NAME="E3:100_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>,<font color="Maroon" title="c5">v2</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" onmouseover="rs('enumset1/D1')" onmouseout="rh()">ENUMSET1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:100_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">V</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:100"><span class="lab"><font color="Green" title="E55">A1</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:100"/>
 <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c6">V</font>,<font color="Maroon" title="c7">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c3">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> (<font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c4">v1</font>) <a href="funct_4.html#K6" title="FUNCT_4:func.6">--&gt;</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">v2</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">a2</font></span>)</span>,<font color="Maroon" title="c1">a1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:100"><span class="lab"><font color="Green" title="E58">A4</font></span></a>, <a class="txt" href="#E5:100"><span class="lab"><font color="Green" title="E59">A5</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E60">A6</font></span></a>, <a class="txt" href="#E2:100"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="txt" href="#E9:100"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="txt" href="#E10:100"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="ref" href="nomin_1.html#T68" target="_self" onmouseover="rs('nomin_1/T68')" onmouseout="rh()">Th68</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
