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

<b>let </b><font color="Maroon">c<sub>1</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>2</sub></font> be  <a href="waybel_3.html#V4">non-Empty</a> <a href="waybel18.html#V1" target="_self">TopSpace-yielding</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>
<i><font color="Green">E8</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:16"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:16"/><i><font color="Green">E10</font></i>: 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:16_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:16_2"/><i><font color="Green">E12</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:16_2"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:16_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:16"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E7:16_2"/><b>then </b><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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="waybel18.html#E1:16"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel18.html#E5:16_2"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E8:16_2"/><i><font color="Green">E15</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:16_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E9:16_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:16_2_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:16_2_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>and </b><br/><a NAME="E5:16_2_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_2_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:16"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:16_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</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="waybel18.html#E1:16"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel18.html#E5:16_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel18.html#E1:16_2_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E4:16_2_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:16_2_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="waybel18.html#E5:16_2_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel18.html#E1:16_2_1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_2_1_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:16_2_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:16"><i><font color="Green">E8</font></i></a>, <a class="txt" href="waybel18.html#E5:16_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel18.html#E2:16_2_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E10:16_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:16_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E11:16_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E12:16_2"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
;<br/>
<a NAME="E13:16_2"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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="waybel18.html#E7:16_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel18.html#E11:16_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<a NAME="E14:16_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E15:16_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:16_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="waybel18.html#E12:16_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:16"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:16_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:16_3"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relat_1.html#K10">"</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E3:16_3"/><b>then </b><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E4:16_3"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:16_3"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E7:16_3"/><i><font color="Green">E14</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:16_3"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:16_3"/><i><font color="Green">E16</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E7:16_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E10:16_3"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> holds <br/><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:16_3_1"/>
<font color="Maroon">c<sub>8</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">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:16_3_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E7:16_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="waybel18.html#E8:16_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="waybel18.html#E9:16_3"><i><font color="Green">E16</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_3_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_1"><b>suppose </b></a><a NAME="E1:16_3_1_1_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K3">proj</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:16_3"><i><font color="Green">E12</font></i></a>, <a class="txt" href="waybel18.html#E6:16_3"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pralg_3.html#D2">PRALG_3:def 2</a></i>;<br/><a NAME="E3:16_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</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="waybel18.html#E2:16_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:16_3_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:16"><i><font color="Green">E9</font></i></a>, <a class="txt" href="waybel18.html#E1:16_3_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_3_1_1_2"><b>suppose </b></a><a NAME="E1:16_3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_3_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:16_3_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:16_3"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:16_3"><i><font color="Green">E13</font></i></a>, <a class="txt" href="waybel18.html#E9:16_3"><i><font color="Green">E16</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" target="_self">proj</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:16"><i><font color="Green">E10</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
