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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="funct_1.html#NM1">Function</a>;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E26</font></i>: 
 <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>thus </b><a NAME="E2:47"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7" target="_self">sproduct</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7" target="_self">sproduct</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="funct_1.html#NM1">Function</a> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E4:47_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E1:47"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ami_1.html#E1:47_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ami_1.html#T29" target="_self">Th29</a></i>;<br/>

<b>thus </b><a NAME="E5:47_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_4.html#T26">FUNCT_4:26</a></i>;<br/>


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

<b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><a NAME="E3:47"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E4:47"/><i><font color="Green">E28</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( 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> <font color="Maroon">c<sub>1</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#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> ) )
 
<b>by </b><i><a class="txt" href="ami_1.html#E3:47"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ami_1.html#E3:47"><i><font color="Green">E27</font></i></a>, <a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>now </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="E1:47_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:47_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E3:47"><i><font color="Green">E27</font></i></a>, <a class="ref" href="grfunc_1.html#T8">GRFUNC_1:8</a></i>;<br/><b>hence </b><a NAME="E3:47_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E4:47"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ami_1.html#E5:47"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ami_1.html#E1:47_2"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:47"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="ami_1.html#K7" target="_self">sproduct</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ami_1.html#E4:47"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ami_1.html#E5:47"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ami_1.html#D16" target="_self">Def16</a></i>;<br/>


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