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<b>let </b><font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c2">A2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">Y1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">A1</font>;<br/><b>let </b><font color="Maroon" title="c4">Y2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c2">A2</font>;<br/><b>let </b><font color="Maroon" title="c5">f</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"><font color="Maroon" title="c1">A1</font>,<font color="Maroon" title="c1">A1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c1">A1</font>;<br/><b>let </b><font color="Maroon" title="c6">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"><font color="Maroon" title="c2">A2</font>,<font color="Maroon" title="c2">A2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c2">A2</font>;<br/><b>let </b><font color="Maroon" title="c7">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c3">Y1</font>;<br/>













<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c3">Y1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c8">G</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<font color="Maroon" title="c4">Y2</font>;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c4">Y2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c9">X</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>;<br/>
<a NAME="E3:4"/><i><font color="Green" title="E5">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="relat_1.html#T89" title="RELAT_1:th.89">RELAT_1:89</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green" title="E6">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green" title="E7">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:4_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:4_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1"/><i><font color="Green" title="E8">A6</font></i>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:4_1_1"/><i><font color="Green" title="E9">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E5:4_1_1"/><i><font color="Green" title="E10">A8</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E6:4_1_1"/><i><font color="Green" title="E11">A9</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E7:4_1_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="relat_1.html#T87" title="RELAT_1:th.87">RELAT_1:87</a></i>;<br/>
<a NAME="E8:4_1_1"/><b>then </b>
( <font color="Maroon" title="c11">x11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c12">x21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c13">x12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> &amp; <font color="Maroon" title="c14">x22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:4_1_1"><i><font color="Green" title="E9">A7</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_1_1"><i><font color="Green" title="E10">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:4_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:4_1_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1_1"><i><font color="Green" title="E11">A9</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:4_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c10">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="E2:4_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E3:4_1"/><b>then </b><i><font color="Green" title="E8">A10</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:4_1"/><b>then </b>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:4_1"/><i><font color="Green" title="E9">A11</font></i>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E7:4_1"/><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="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E8:4_1"/><i><font color="Green" title="E11">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E9:4_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_1"><i><font color="Green" title="E8">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:4_1"/><b>then </b>
( <font color="Maroon" title="c11">x11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c13">x12</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> &amp; <font color="Maroon" title="c12">x21</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y1</font> &amp; <font color="Maroon" title="c14">x22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y2</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:4_1"/><b>then </b><i><font color="Green" title="E12">A14</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</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"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:4_1"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c3">Y1</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">g</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c4">Y2</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>;<br/>
<a NAME="E14:4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E7:4_1"><i><font color="Green" title="E10">A12</font></i></a>, <a class="txt" href="nattra_1.html#E8:4_1"><i><font color="Green" title="E11">A13</font></i></a>, <a class="txt" href="nattra_1.html#E12:4_1"><i><font color="Green" title="E12">A14</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:4_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1"><i><font color="Green" title="E9">A11</font></i></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E8">A15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green" title="E9">A16</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 ;<br/><b>then consider </b><font color="Maroon" title="c11">x11</font>, <font color="Maroon" title="c12">x21</font>, <font color="Maroon" title="c13">x12</font>, <font color="Maroon" title="c14">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green" title="E10">A17</font></i>: 
<font color="Maroon" title="c10">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>and </b><br/><a NAME="E4:4_2"/><i><font color="Green" title="E11">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">F</font>
 <b>and </b><br/><a NAME="E5:4_2"/><i><font color="Green" title="E12">A19</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">G</font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>;<br/><b>thus </b><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_2"><i><font color="Green" title="E10">A17</font></i></a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green" title="E11">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green" title="E12">A19</font></i></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green" title="E11">A18</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c12">x21</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x12</font>,<font color="Maroon" title="c14">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:4"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green" title="E12">A19</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x11</font>,<font color="Maroon" title="c13">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x21</font>,<font color="Maroon" title="c14">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green" title="E11">A18</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green" title="E12">A19</font></i></a>, <a class="ref" href="funct_4.html#D3" title="FUNCT_4:def.3">FUNCT_4:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="nattra_1.html#E1:4_2"><i><font color="Green" title="E9">A16</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_2"><i><font color="Green" title="E10">A17</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:4"/><i><font color="Green" title="E9">A20</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">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="E1:4_3"/>
<font color="Maroon" title="c10">x</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> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_3"/><i><font color="Green" title="E10">A21</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:4_3"/><i><font color="Green" title="E11">A22</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:4_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="nattra_1.html#E6:4"><i><font color="Green" title="E8">A15</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_3"><i><font color="Green" title="E10">A21</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_3"><i><font color="Green" title="E11">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:4_3"/>
<font color="Maroon" title="c10">x</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> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_3"><i><font color="Green" title="E10">A21</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:4"/><i><font color="Green" title="E10">A23</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> holds <br/><span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <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"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4"><i><font color="Green" title="E8">A15</font></i></a></i>;<br/>
<a NAME="E9:4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:4"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E7:4"><i><font color="Green" title="E9">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:4"/><b>then </b>
<span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> is   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>,<span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:4"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="nattra_1.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E12:4"/>
<span class="p1"><a href="nattra_1.html#K2" title="NATTRA_1:func.2">|:</a><span class="default"><font color="Maroon" title="c7">F</font>,<font color="Maroon" title="c8">G</font></span><a href="nattra_1.html#K2" title="NATTRA_1:func.2">:|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="cat_2.html#K10" title="CAT_2:func.10">|:</a><span class="default"><font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font></span><a href="cat_2.html#K10" title="CAT_2:func.10">:|</a></span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c3">Y1</font>,<font color="Maroon" title="c4">Y2</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="nattra_1.html#E8:4"><i><font color="Green" title="E10">A23</font></i></a>, <a class="ref" href="partfun1.html#T34" title="PARTFUN1:th.34">PARTFUN1:34</a></i>;<br/>


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