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<div class="add">

<b>set </b><font color="Maroon" title="c6">EE</font> =  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>;<br/>
<b>set </b><font color="Maroon" title="c7">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> : verum </span>}</span> ;<br/>
<div><i><font color="Green" title="E7">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y1</font>, <font color="Maroon" title="c10">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c11">X1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E3:14_1_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c11">X1</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><b>assume </b><a NAME="E4:14_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c12">X2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E6:14_1_1_1"/><i><font color="Green" title="E9">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c10">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c12">X2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">X2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E7:14_1_1_1"/><i><font color="Green" title="E10">A5</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X1</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X1</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E3:14_1_1_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E8:14_1_1_1"/>
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">X2</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c12">X2</font> )
 <b>by </b><i><a class="txt" href="group_10.html#E6:14_1_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E9:14_1_1_1"/>
<font color="Maroon" title="c9">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:14_1_1_1"><i><font color="Green" title="E10">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon" title="c9">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E3:14_1_1_2"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c8">x</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="c9">X</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><b>reconsider </b><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">X</font>, <font color="Maroon" title="c11">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c9">X</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c10">y</font>;<br/><b>take </b><font color="Maroon" title="c13">z</font> = <font color="Maroon" title="c11">z</font>;<br/><b>thus </b><a NAME="E5:14_1_1_2"/>
<font color="Maroon" title="c8">x</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="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:14_1_1_2"><i><font color="Green" title="E8">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> : verum </span>}</span>  as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="group_10.html#E1:14_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#D1" title="FUNCT_1:def.1">FUNCT_1:def 1</a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:14_1_1_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">X</font> = <font color="Maroon" title="c9">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> ;<br/><b>reconsider </b><font color="Maroon" title="c11">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">X</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/><b>take </b><font color="Maroon" title="c12">y</font> = <font color="Maroon" title="c11">y</font>;<br/><b>thus </b><a NAME="E4:14_1_1_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font>
 ;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E3:14_1_1_3"/><i><font color="Green" title="E8">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">X</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E5:14_1_1_3"/><i><font color="Green" title="E9">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c11">X</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:14_1_1_3"><i><font color="Green" title="E8">A7</font></i></a></i>;<br/><a NAME="E6:14_1_1_3"/><i><font color="Green" title="E10">A9</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:14_1_1_3"><i><font color="Green" title="E9">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E7:14_1_1_3"/>
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="group_10.html#E1:14"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="group_10.html#T2" target="_self" title="GROUP_10:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E8:14_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="group_10.html#E6:14_1_1_3"><i><font color="Green" title="E10">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:14_1_1"/><b>then </b><i><font color="Green" title="E8">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_1_4"/>
<font color="Maroon" title="c9">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="c8">f</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c10">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:14_1_1_4"/><i><font color="Green" title="E9">A11</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c9">Y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5" title="RELAT_1:def.5">RELAT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">X2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E5:14_1_1_4"/><i><font color="Green" title="E10">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">X1</font>,<font color="Maroon" title="c9">Y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c11">X2</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X2</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="group_10.html#E3:14_1_1_4"><i><font color="Green" title="E9">A11</font></i></a></i>;<br/><a NAME="E6:14_1_1_4"/>
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="group_10.html#E1:14"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="group_10.html#T2" target="_self" title="GROUP_10:th.2">Th2</a></i>;<br/><a NAME="E7:14_1_1_4"/><b>then </b><i><font color="Green" title="E11">A13</font></i>: 
<font color="Maroon" title="c11">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 ;<br/><a NAME="E8:14_1_1_4"/><b>then </b><i><font color="Green" title="E12">A14</font></i>: 
 ex <font color="Olive" title="b1">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">E</font> st <br/>( <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X2</font> &amp;  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> )
 ;<br/><a NAME="E9:14_1_1_4"/><i><font color="Green" title="E13">A15</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E5:14_1_1_4"><i><font color="Green" title="E10">A12</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E10:14_1_1_4"/>
<font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c1">E</font>,<font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T12" title="FUNCT_2:th.12">FUNCT_2:12</a></i>;<br/><a NAME="E11:14_1_1_4"/><b>then </b><i><font color="Green" title="E14">A16</font></i>: 
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">E</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>set </b><font color="Maroon" title="c12">fe</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">X2</font>;<br/><a NAME="E12:14_1_1_4"/>
<font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="group_10.html#T1" target="_self" title="GROUP_10:th.1">Th1</a></i>;<br/><a NAME="E13:14_1_1_4"/><b>then </b><i><font color="Green" title="E15">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">X2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/><a NAME="E14:14_1_1_4"/><i><font color="Green" title="E16">A18</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X2</font>
 <b>by </b><i><a class="txt" href="group_10.html#E7:14_1_1_4"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="group_10.html#E11:14_1_1_4"><i><font color="Green" title="E14">A16</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/><a NAME="E15:14_1_1_4"/><i><font color="Green" title="E17">A19</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c11">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y</font>
 <b>by </b><i><a class="txt" href="group_10.html#E9:14_1_1_4"><i><font color="Green" title="E13">A15</font></i></a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">n'</font> = <font color="Maroon" title="c2">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E17:14_1_1_4"/>
<font color="Maroon" title="c11">X2</font>,<font color="Maroon" title="c9">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">are_equipotent</a> 
 <b>by </b><i><a class="txt" href="group_10.html#E13:14_1_1_4"><i><font color="Green" title="E15">A17</font></i></a>, <a class="txt" href="group_10.html#E14:14_1_1_4"><i><font color="Green" title="E16">A18</font></i></a>, <a class="txt" href="group_10.html#E15:14_1_1_4"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a></i>;<br/><a NAME="E18:14_1_1_4"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="group_10.html#E8:14_1_1_4"><i><font color="Green" title="E12">A14</font></i></a>, <a class="ref" href="card_1.html#T21" title="CARD_1:th.21">CARD_1:21</a></i>;<br/><b>hence </b><a NAME="E19:14_1_1_4"/>
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 <b>by </b><i><a class="txt" href="group_10.html#E9:14_1_1_4"><i><font color="Green" title="E13">A15</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:14_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>, <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> <b>by </b><i><a class="txt" href="group_10.html#E5:14_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">f</font>
;<br/>

<b>thus </b><a NAME="E9:14_1_1"/>
for <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> holds  <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">X</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font>;<br/>

<a NAME="E1:14_1_1_5"/>
not  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="group_10.html#E1:14"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="group_10.html#T2" target="_self" title="GROUP_10:th.2">Th2</a></i>;<br/>
<a NAME="E2:14_1_1_5"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">f</font>
 
<b>by </b><i><a class="txt" href="group_10.html#E5:14_1_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">X'</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="group_10.html#K3" title="GROUP_10:func.3">the_subsets_of_card</a> <font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">E</font><b> such that </b><br/><a NAME="E4:14_1_1_5"/><i><font color="Green" title="E9">A20</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">X</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c11">X'</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/>
<a NAME="E5:14_1_1_5"/>
( <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X'</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c11">X'</font> )
 
<b>by </b><i><a class="txt" href="group_10.html#E4:14_1_1_5"><i><font color="Green" title="E9">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1_1_5"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">LO</font> <a href="group_10.html#K1" title="GROUP_10:func.1">^</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">X</font>
 ;<br/>


</div><b>end;</b></div>


</div>
