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

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:30_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>
<a NAME="E2:30_1_2"/><i><font color="Green">E26</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_2.html#K6">pr2</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="funct_2.html#K6">pr2</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E2:30_1_2_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E1:30_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure7.html#E2:30_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="measure7.html#E1:30_1_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/><a NAME="E3:30_1_2_1"/>
<span class="p1">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="measure7.html#E1:30_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure7.html#E2:30_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="mcart_1.html#D12">MCART_1:def 12</a></i>;<br/><b>hence </b><a NAME="E4:30_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="measure7.html#E2:30_1_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:30_1_2"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>

<a NAME="E5:30_1_2"/><i><font color="Green">E28</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="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <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"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E3:30_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="measure7.html#K12" target="_self">pr1</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="txt" href="measure7.html#E4:30_1_2"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="mcart_1.html#K2">`2</a> 
 <b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:30_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="measure7.html#E1:30_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="measure7.html#E2:30_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a>  <a href="funct_2.html#K6">pr2</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure7.html#E5:30_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="mcart_1.html#D13">MCART_1:def 13</a></i>;<br/>


</div>
