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<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">D</font> be  <a href="comput_1.html#V1" title="COMPUT_1:attr.1">compatible</a> <a href="rfunct_3.html#NM2" title="RFUNCT_3:NM.2">PFUNC_DOMAIN</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>;<br/>





<b>set </b><font color="Maroon" title="c4">E</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c5">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">u</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">D</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/>
<a NAME="E2:22"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">u</font>
 ;<br/>

<a NAME="E2:22_1"/><b>then </b>
<font color="Maroon" title="c7">x</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T15" target="_self" title="COMPUT_1:th.15">Th15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:22_1"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c8">Z</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="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">f</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font><b> such that </b><br/><a NAME="E6:22_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:22_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:22_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:22_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="comput_1.html#E6:22_1"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:22"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">u</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon" title="c7">y</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="c6">u</font>
 ;<br/>

<a NAME="E2:22_2"/><b>then </b>
<font color="Maroon" title="c7">y</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="comput_1.html#T17" target="_self" title="COMPUT_1:th.17">Th17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">Z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:22_2"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c8">Z</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="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font></span>)</span> where <font color="Olive" title="b1">f</font> is    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font> : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">f</font> being    <a href="rfunct_3.html#M2" title="RFUNCT_3:mode.2">Element</a> of <font color="Maroon" title="c3">D</font><b> such that </b><br/><a NAME="E6:22_2"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:22_2"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E7:22_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Y</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:22_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E4:22_2"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="comput_1.html#E6:22_2"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:22"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c3">D</font> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="comput_1.html#E2:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>


</div>
