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<b>let </b><font color="Maroon" title="c3">IT</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1_2_1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">X</font>
 ;<br/><b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E2:5_1_2_1"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E3:5_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">i</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="c2">X</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E4:5_1_2_1"/><b>then </b><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</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"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="msafree.html#E1:5_1_2_1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="card_3.html#D3" title="CARD_3:def.3">CARD_3:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">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="E1:5_1_2_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:5_1_2_1_1_1"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>and </b><br/><a NAME="E4:5_1_2_1_1_1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c5">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="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="msafree.html#E4:5_1_2_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>take </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a</font>;<br/><b>thus </b><a NAME="E5:5_1_2_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="msafree.html#E3:5_1_2_1_1_1"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/><b>thus </b><a NAME="E6:5_1_2_1_1_1"/>
<font color="Maroon" title="c5">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="c8">a</font>,<font color="Maroon" title="c4">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="msafree.html#E3:5_1_2_1_1_1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="msafree.html#E4:5_1_2_1_1_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>given </b><font color="Maroon" title="c6">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:5_1_2_1_1"/><i><font color="Green" title="E7">A6</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E3:5_1_2_1_1"/><i><font color="Green" title="E8">A7</font></i>: 
<font color="Maroon" title="c5">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="c6">a</font>,<font color="Maroon" title="c4">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><a NAME="E4:5_1_2_1_1"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">i</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:5_1_2_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="txt" href="msafree.html#E4:5_1_2_1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="msafree.html#E2:5_1_2_1_1"><i><font color="Green" title="E7">A6</font></i></a>, <a class="txt" href="msafree.html#E3:5_1_2_1_1"><i><font color="Green" title="E8">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:5_1_2_1"/>
<font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="msafree.html#E2:5_1_2_1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="msafree.html#D2" target="_self" title="MSAFREE:def.2">Def2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:5_1_2"/><i><font color="Green" title="E4">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E3:5_1_2"/><i><font color="Green" title="E5">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:5_1_2"/><b>then </b><i><font color="Green" title="E6">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:5_1_2_2"/><i><font color="Green" title="E7">A11</font></i>: 
<font color="Maroon" title="c4">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="c2">X</font>
 ;<br/><a NAME="E2:5_1_2_2"/><b>then </b><i><font color="Green" title="E8">A12</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><div><i><font color="Green" title="E9">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</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="E1:5_1_2_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:5_1_2_2_1_1"/><i><font color="Green" title="E10">A14</font></i>: 
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
 <b>and </b><br/><a NAME="E4:5_1_2_2_1_1"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c5">a</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="c6">b</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="msafree.html#E2:5_1_2_2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="ref" href="msafree.html#D2" target="_self" title="MSAFREE:def.2">Def2</a></i>;<br/><a NAME="E5:5_1_2_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:5_1_2_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="msafree.html#E3:5_1_2_2_1_1"><i><font color="Green" title="E10">A14</font></i></a>, <a class="txt" href="msafree.html#E4:5_1_2_2_1_1"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:5_1_2_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a1</font>, <font color="Maroon" title="c7">a2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:5_1_2_2_1"/><i><font color="Green" title="E10">A16</font></i>: 
( <font color="Maroon" title="c6">a1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c7">a2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>and </b><br/><a NAME="E5:5_1_2_2_1"/><i><font color="Green" title="E11">A17</font></i>: 
<font color="Maroon" title="c5">a</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="c6">a1</font>,<font color="Maroon" title="c7">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><a NAME="E6:5_1_2_2_1"/>
<font color="Maroon" title="c7">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="msafree.html#E4:5_1_2_2_1"><i><font color="Green" title="E10">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E7:5_1_2_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="msafree.html#E2:5_1_2_2"><i><font color="Green" title="E8">A12</font></i></a>, <a class="txt" href="msafree.html#E4:5_1_2_2_1"><i><font color="Green" title="E10">A16</font></i></a>, <a class="txt" href="msafree.html#E5:5_1_2_2_1"><i><font color="Green" title="E11">A17</font></i></a>, <a class="ref" href="msafree.html#D2" target="_self" title="MSAFREE:def.2">Def2</a></i>;<br/></div><b>end;</b></div><b>thus </b><font color="Maroon" title="c3">IT</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
 <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="txt" href="msafree.html#E2:5_1_2"><i><font color="Green" title="E4">A8</font></i></a>, <a class="txt" href="msafree.html#E3:5_1_2"><i><font color="Green" title="E5">A9</font></i></a>, <a class="txt" href="msafree.html#E4:5_1_2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="msafree.html#E1:5_1_2_2"><i><font color="Green" title="E7">A11</font></i></a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="txt" href="msafree.html#E3:5_1_2_2"><i><font color="Green" title="E9">A13</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:5_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K2" title="CARD_3:func.2">disjoin</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="msafree.html#E4:5_1_2"><i><font color="Green" title="E6">A10</font></i></a>, <a class="ref" href="card_3.html#D3" title="CARD_3:def.3">CARD_3:def 3</a></i>;<br/>


</div>
