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<span class="kw">let </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</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"> (  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; (  for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>) ) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; (  for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>) ) implies <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:103_1_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E2:103_1_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</font>)
 <span class="kw">and </span><br/><a NAME="E3:103_1_2"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">and </span><br/><a NAME="E4:103_1_2"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
 for <font color="Olive" title="b1">d1</font> being    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font><br/>  for <font color="Olive" title="b2">d2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st  not <font color="Olive" title="b1">d1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) holds <br/><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>,<font color="Maroon" title="c3">v</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"> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font></span><br/>

<span class="kw">thus </span><a NAME="E5:103_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E3:103_1_2"><span class="lab"><font color="Green" title="E57">A11</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D11" onmouseover="rs('funct_1/D11')" onmouseout="rh()">FUNCT_1:def 11</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">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">D</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"> (  not <font color="Maroon" title="c6">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">f</font> or <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:103_1_2"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
<font color="Maroon" title="c6">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</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"> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">D1</font>, <font color="Maroon" title="c8">D2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:103_1_2"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
( <font color="Maroon" title="c7">D1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>)</span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">D2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nomin_1.html#K3" title="NOMIN_1:func.3">ND</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>) )
 <span class="kw">and </span><br/><a NAME="E9:103_1_2"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c6">D</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="c7">D1</font>,<font color="Maroon" title="c8">D2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:103_1_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E6:103_1_2"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">D1</font> = <font color="Maroon" title="c7">D1</font> as    <a href="nomin_1.html#M3" title="NOMIN_1:mode.3">NonatomicND</a> of <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:103_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="nomin_1.html#T43" target="_self" onmouseover="rs('nomin_1/T43')" onmouseout="rh()">Th43</a></span>;<br/>
<a NAME="E11:103_1_2"/><span class="lab"><font color="Green" title="E62">A16</font></span>: 
 not <font color="Maroon" title="c9">D1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:103_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font> = 
 <a href="nomin_1.html#K17" title="NOMIN_1:func.17">local_overlapping</a> (<font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c9">D1</font>,<font color="Maroon" title="c8">D2</font>,<font color="Maroon" title="c3">v</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:103_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E8:103_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="txt" href="#E9:103_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">D</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:103_1_2"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E8:103_1_2"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="txt" href="#E9:103_1_2"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E11:103_1_2"><span class="lab"><font color="Green" title="E62">A16</font></span></a></span>
;<br/> <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/>


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