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<div class="add">

<b>let </b><font color="Maroon" title="c1">S</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="amistd_2.html#V1" title="AMISTD_2:attr.1">product-like</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:13"/>
<font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="amistd_2.html#T9" target="_self" title="AMISTD_2:th.9">Th9</a></i>; <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><br/>

<b>let </b><font color="Maroon" title="c2">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="E2:13"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">g</font>
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green" title="E12">A2</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span>
 <b>and </b><br/><a NAME="E6:13"/><i><font color="Green" title="E13">A3</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">p</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E8:13"/><i><font color="Green" title="E14">A4</font></i>: 
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="amistd_2.html#D2" target="_self" title="AMISTD_2:def.2">Def2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">s</font> being    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of <font color="Maroon" title="c1">S</font>;<br/>
<i><font color="Green" title="E15">A5</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">g</font> = 
 <a href="pralg_2.html#K1" title="PRALG_2:func.1">DOM</a> <font color="Maroon" title="c1">S</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E5:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="amistd_2.html#T7" target="_self" title="AMISTD_2:th.7">Th7</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">s</font>
<b>by </b><i><a class="ref" href="pralg_2.html#D2" title="PRALG_2:def.2">PRALG_2:def 2</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font>
<b>by </b><i><a class="txt" href="amistd_2.html#E8:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
;<br/>
<a NAME="E11:13"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green" title="E16">A6</font></i>: 
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">p</font>
 ;<br/>

<a NAME="E2:13_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K1" title="AMISTD_2:func.1">PA</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E5:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="amistd_2.html#E6:13"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="amistd_2.html#E10:13"><i><font color="Green" title="E15">A5</font></i></a></i>;<br/>
<a NAME="E3:13_2"/><b>then </b>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K5" title="CARD_3:func.5">pi</a> <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c6">z</font>
 
<b>by </b><i><a class="txt" href="amistd_2.html#E5:13"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="amistd_2.html#E10:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="amistd_2.html#E1:13_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="amistd_2.html#D1" target="_self" title="AMISTD_2:def.1">Def1</a></i>;<br/>
<a NAME="E4:13_2"/><b>then </b>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D6" title="CARD_3:def.6">CARD_3:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:13_2"/>
<font color="Maroon" title="c3">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E8:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="amistd_2.html#E1:13_2"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:13"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E4:13"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="amistd_2.html#E8:13"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="amistd_2.html#E10:13"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>


</div>
