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

<b>let </b><font color="Maroon">I</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be  <a href="group_7.html#V2" target="_self">Group-like</a> <a href="group_7.html#V3" target="_self">associative</a> <a href="group_7.html#NM1">HGrStr-Family</a> of <font color="Maroon">I</font>;<br/>



<b>set </b><font color="Maroon">GP</font> =  <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="group_7.html#K3" target="_self">sum</a> <font color="Maroon">F</font>;<br/>
<a NAME="E1:30"/><i><font color="Green">E29</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K3" target="_self">sum</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:30_1"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K3" target="_self">sum</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>reconsider </b><font color="Maroon">g</font> =  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> as    <a href="pralg_2.html#M1">Element</a> of  <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">F</font></span>)</span> <b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">x</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="E3:30_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as    <a href="pralg_2.html#M1">Element</a> of  <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">F</font></span>)</span> <b>by </b><i/>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="pralg_2.html#M1">Element</a> of  <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">F</font></span>)</span>;<br/><b>thus </b> <a href="relat_1.html#K1">dom</a> <font color="Maroon">p</font> = 
 <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">F</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:30_1"/>
 ex <font color="Olive">g</font> being   <a href="pralg_2.html#M1">Element</a> of  <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">F</font></span>)</span> ex <font color="Olive">J</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ex <font color="Olive">f</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Olive">J</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> <a href="funct_4.html#K1">+*</a> <font color="Olive">f</font> &amp; ( for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font> holds <br/> ex <font color="Olive">G</font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">G</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">G</font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">m</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">G</font> st <br/>( <font color="Olive">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> );<br/>
<b>consider </b><font color="Maroon">J</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:30_1_2"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> iff ( <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:30_1_2"/>
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</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="E1:30_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>hence </b><a NAME="E2:30_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">ff</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E6:30_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/><font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4">PBOOLE:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">ff</font>
;<br/>

<b>thus </b><a NAME="E7:30_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<a NAME="E8:30_1_2"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:30_1_2"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E57</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">ff</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">ff</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:30_1_2_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_3_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_3_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_3_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">ff</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_3_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_3_1_2"/><i><font color="Green">E60</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">G</font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a> <b> such that </b><br/><a NAME="E3:30_1_2_3_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:30_1_2_3_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">ff</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E12:30_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">ff</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>

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

<b>assume </b><a NAME="E13:30_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon">G</font> being  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V3">Group-like</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">m</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E15:30_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E16:30_1_2"/>
<font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E17:30_1_2"/>
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">ff</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:30_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K3" target="_self">sum</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/>
 <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T63">GROUP_2:63</a></i>;<br/>
<b>hence </b><a NAME="E3:30"/>
 <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="group_7.html#K3" target="_self">sum</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T68">GROUP_2:68</a></i>;<br/>


</div>
