<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<font color="Maroon">c<sub>3</sub></font>;<br/>













<b>assume </b><a NAME="E1:4"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>assume </b><a NAME="E2:4"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="realset1.html#K1">||</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span>;<br/>
<a NAME="E3:4"/><i><font color="Green">E5</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E4:4"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E7</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</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#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_1_1"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:4_1_1"/><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:4_1_1"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E6:4_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E7:4_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E8:4_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:4_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:4_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:4_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E12:4_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
 ;<br/>

<a NAME="E3:4_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E4:4_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:4_1"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E7:4_1"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E8:4_1"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>;<br/>
<a NAME="E9:4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E11:4_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E12:4_1"/><b>then </b><i><font color="Green">E12</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:4_1"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="mcart_1.html#K12">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="nattra_1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E14:4_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="nattra_1.html#E7:4_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="nattra_1.html#E8:4_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E12:4_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:4_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E6:4_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E8</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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:4_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E4:4_2"/><i><font color="Green">E11</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E5:4_2"/><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>;<br/><b>thus </b><span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E1:4"><i><font color="Green">E3</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span><a href="tarski.html#K4">]</a></span></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="nattra_1.html#E2:4"><i><font color="Green">E4</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E3:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="nattra_1.html#E5:4_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_4.html#D3">FUNCT_4:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="nattra_1.html#E1:4_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:4"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</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">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:4_3"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:4_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:4_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="nattra_1.html#E6:4"><i><font color="Green">E8</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="nattra_1.html#E4:4_3"><i><font color="Green">E11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:4_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="nattra_1.html#E3:4_3"><i><font color="Green">E10</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:4"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> holds <br/><span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E6:4"><i><font color="Green">E8</font></i></a></i>;<br/>
<a NAME="E9:4"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E10:4"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E7:4"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:4"/><b>then </b>
<span class="p1"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span> is   <a href="partfun1.html#NM1">PartFunc</a> of <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p2"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span></span><a href="mcart_1.html#K12">:]</a></span>,<span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="txt" href="nattra_1.html#E4:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="nattra_1.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>hence </b><a NAME="E12:4"/>
<span class="p1"><a href="nattra_1.html#K2" target="_self">|:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="nattra_1.html#K2" target="_self">:|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="cat_2.html#K10">|:</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="cat_2.html#K10">:|</a></span> <a href="realset1.html#K1">||</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="mcart_1.html#K12">:]</a></span>
 <b>by </b><i><a class="txt" href="nattra_1.html#E5:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="nattra_1.html#E8:4"><i><font color="Green">E10</font></i></a>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
