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

<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">CF</font> =  <a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font>;<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/>
<a NAME="E2:28_1_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</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">F</font></span>)</span> &amp;  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T18">CARD_3:18</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_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>
<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">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="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">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> ) ) );<br/>
<b>consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:28_1_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> iff ( <font color="Olive">x</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">F</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:28_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:28_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:28_1_1_2"/><b>then </b>
<font color="Maroon">a</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">F</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:28_1_1_2"/>
<font color="Maroon">a</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>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:28_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">J</font> =  <a href="xboole_0.html#K1">{}</a>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:28_1_1_3"/>
 <a href="finseq_1.html#K4">dom</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">J</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

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

<b>thus </b><a NAME="E4:28_1_1_3"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22">FUNCT_4:22</a></i>;<br/>

<b>thus </b><a NAME="E5:28_1_1_3"/>
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/> 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="Maroon">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="Maroon">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> )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">b</font> = the <a href="group_1.html#U1">mult</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> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font>;<br/>
<a NAME="E9:28_1_1"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:28_1_1_4"/>
 <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:28_1_1_4"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> the <a href="group_1.html#U1">mult</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="zfmisc_1.html#T119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E3:28_1_1_4"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon">y</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="E4:28_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:28_1_1_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="group_1.html#U1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:28_1_1_4"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">J1</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>, <font color="Maroon">f1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J1</font><b> such that </b><br/><a NAME="E10:28_1_1_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f1</font>
 <b>and </b><br/><a NAME="E11:28_1_1_4"/><i><font color="Green">E59</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">J1</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="Maroon">f1</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="Maroon">f1</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> )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">J2</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>, <font color="Maroon">f2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J2</font><b> such that </b><br/><a NAME="E13:28_1_1_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f2</font>
 <b>and </b><br/><a NAME="E14:28_1_1_4"/><i><font color="Green">E61</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">J2</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="Maroon">f2</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="Maroon">f2</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> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E15:28_1_1_4"/><i><font color="Green">E63</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J1</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E64</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x1</font> = 
<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">f1</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="28_1_1_4_2"><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="E1:28_1_1_4_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:28_1_1_4_2"/><b>then </b><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</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="28_1_1_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_4_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_2_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_4_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_2_1_1"/><b>then </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>  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">i</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_4_2_1_1"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_2_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_4_2_1_2"/><i><font color="Green">E68</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</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="E3:28_1_1_4_2_1_2"/><i><font color="Green">E71</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><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, , </i>
;<br/><b>hence </b><a NAME="E5:28_1_1_4_2_1_2"/>
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</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>, <a class="ref" href="group_7.html#T1" target="_self">Th1</a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<i><font color="Green">E72</font></i>:  <a href="relat_1.html#K1">dom</a> <font color="Maroon">x2</font> = 
<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">f2</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="28_1_1_4_4"><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="E1:28_1_1_4_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:28_1_1_4_4"/><b>then </b><i><font color="Green">E75</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</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="28_1_1_4_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_4_4_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_4_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_4_4_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_4_1_1"/><b>then </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>  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">i</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_4_4_1_1"/>
<font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_4_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_4_4_1_2"/><i><font color="Green">E97</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</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="E3:28_1_1_4_4_1_2"/><i><font color="Green">E98</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/><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, , </i>
;<br/><b>hence </b><a NAME="E5:28_1_1_4_4_1_2"/>
<font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x2</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>, <a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">x1</font>, <font color="Maroon">X2</font> = <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</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/>;<br/>
<b>reconsider </b><font color="Maroon">ff</font> = <font color="Maroon">X1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">X2</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/>
<b>defpred </b><font color="Maroon">S<sub>2</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">k1</font>, <font color="Olive">k2</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">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">k1</font> <a href="group_1.html#K1">*</a> <font color="Olive">k2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">G</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></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">K</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E26:28_1_1_4"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">k</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> iff ( <font color="Olive">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green">E100</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_5"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">J</font> =  <a href="xboole_0.html#K1">{}</a>  as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>take </b><font color="Maroon">J</font> = <font color="Maroon">J</font>;<br/><b>thus </b><a NAME="E2:28_1_1_4_5"/>
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/> 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="Maroon">ff</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="Maroon">ff</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> )
 ;<br/></div><b>end;</b></div>
<a NAME="E28:28_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:28_1_1_4"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y1</font> = <font color="Maroon">y</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/>
<a NAME="E31:28_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_6"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">J</font> = <span class="p1">(<span class="default"><font color="Maroon">J1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">K</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b>
<font color="Maroon">J</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <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">k1</font>, <font color="Olive">k2</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">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">k1</font> <a href="group_1.html#K1">*</a> <font color="Olive">k2</font> );<br/>
<a NAME="E2:28_1_1_4_6"/><i><font color="Green">E102</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/> ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Olive">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:28_1_1_4_6_1"/><i><font color="Green">E104</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> <b> such that </b><br/><a NAME="E3:28_1_1_4_6_1"/><i><font color="Green">E105</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>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>, <font color="Maroon">k2</font> = <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">k1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">k2</font>
;<br/>

<b>thus </b><a NAME="E5:28_1_1_4_6_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">j</font>,<font color="Maroon">k1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">k2</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E4:28_1_1_4_6"/><i><font color="Green">E106</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">S<sub>3</sub></font>[<font color="Olive">j</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>set </b><font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>;<br/>
<a NAME="E5:28_1_1_4_6"/><i><font color="Green">E108</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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="E6:28_1_1_4_6"/><i><font color="Green">E109</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E110</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">f</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">f</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="28_1_1_4_6_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="E1:28_1_1_4_6_3"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:28_1_1_4_6_3"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</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> <a href="binop_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Fi</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">FF</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="E5:28_1_1_4_6_3"/><i><font color="Green">E177</font></i>: 
<font color="Maroon">FF</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/><b>reconsider </b><font color="Maroon">Fi</font> = <font color="Maroon">Fi</font> as  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>by </b><i>, <a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">x1i</font> = <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>, <font color="Maroon">x2i</font> = <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> <b>by </b><i>, , </i>;<br/><a NAME="E8:28_1_1_4_6_3"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1i</font> <a href="group_1.html#K1">*</a> <font color="Maroon">x2i</font>
 <b>by </b><i>, </i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_4_6_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="28_1_1_4_6_3_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_4_6_3_1_1"/><i><font color="Green">E181</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_3_1_1"/><b>then </b>
 ex <font color="Olive">GG</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">k1a</font>, <font color="Olive">k2a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">GG</font> st <br/>( <font color="Olive">GG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">k1a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">k2a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">k1a</font> <a href="group_1.html#K1">*</a> <font color="Olive">k2a</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_4_6_3_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , <a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_3_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_4_6_3_1_2"/><i><font color="Green">E185</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>then </b><i><font color="Green">E186</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">FF</font>
<b>by </b><i>, <a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a>, </i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4_6_3_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_3_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_4_6_3_1_2_2_1"/>
(  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J2</font> or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 </span><b>by </b><i><a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_3_1_2_2_1_1"><b>case </b></a><a NAME="E1:28_1_1_4_6_3_1_2_2_1_1"/>
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J2</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:28_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <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/><a NAME="E4:28_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">FF</font> )
 <b>by </b><i>, <a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a>, </i>;<br/><b>hence </b><a NAME="E5:28_1_1_4_6_3_1_2_2_1_1"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_3_1_2_2_1_2"><b>case </b></a><a NAME="E1:28_1_1_4_6_3_1_2_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_3_1_2_2_1_2"/><b>then </b>
 ex <font color="Olive">GG</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">k1</font>, <font color="Olive">k2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">GG</font> st <br/>( <font color="Olive">GG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">k1</font> <a href="group_1.html#K1">*</a> <font color="Olive">k2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">GG</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">GG</font> &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">GG</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_4_6_3_1_2_2_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:28_1_1_4_6_3_1_2"/>
<font color="Maroon">y1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:28_1_1_4_6"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self">Def1</a>, , <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="E10:28_1_1_4_6"/><i><font color="Green">E187</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">k1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">k2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E12:28_1_1_4_6"/><i><font color="Green">E188</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">k1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">k2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">k2</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E13:28_1_1_4_6"/>
<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>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E14:28_1_1_4_6"/>
<font color="Maroon">f</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>
 <b>by </b><i/>;<br/>

<a NAME="E15:28_1_1_4_6"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">J2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_4_6_4"/>
( ( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> ) or ( not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> ) or ( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> ) )
 </span><b>by </b><i><a class="txt" href="group_7.html#E3:28_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_4_1"><b>suppose </b></a><a NAME="E1:28_1_1_4_6_4_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_4_1"/><b>then </b><i><font color="Green">E191</font></i>: 
<font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>consider </b><font color="Maroon">G1</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="E4:28_1_1_4_6_4_1"/><i><font color="Green">E193</font></i>: 
( <font color="Maroon">G1</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">f1</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">G1</font> &amp; <font color="Maroon">f1</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">G1</font> )
 <b>by </b><i>, </i>;<br/><font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G1</font>
<b>by </b><i>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, </i>
;<br/><b>hence </b><a NAME="E6:28_1_1_4_6_4_1"/>
<font color="Maroon">f</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>, <a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_4_2"><b>suppose </b></a><a NAME="E1:28_1_1_4_6_4_2"/><i><font color="Green">E194</font></i>: 
( not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_4_2"/><b>then </b><i><font color="Green">E195</font></i>: 
<font color="Maroon">x2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>consider </b><font color="Maroon">G2</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="E4:28_1_1_4_6_4_2"/><i><font color="Green">E197</font></i>: 
( <font color="Maroon">G2</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">f2</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">G2</font> &amp; <font color="Maroon">f2</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">G2</font> )
 <b>by </b><i>, </i>;<br/><font color="Maroon">x1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G2</font>
<b>by </b><i>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, </i>
;<br/><b>hence </b><a NAME="E6:28_1_1_4_6_4_2"/>
<font color="Maroon">f</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>, <a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a>, <a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_4_6_4_3"><b>suppose </b></a><a NAME="E1:28_1_1_4_6_4_3"/><i><font color="Green">E198</font></i>: 
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J1</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J2</font> )
 ;<br/><div class="add"><a NAME="E2:28_1_1_4_6_4_3"/><b>then </b><i><font color="Green">E199</font></i>: 
 ex <font color="Olive">G1</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">G1</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">f1</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="Olive">G1</font> &amp; <font color="Maroon">f1</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="Olive">G1</font> )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_1_4_6_4_3"/><i><font color="Green">E200</font></i>: 
 ex <font color="Olive">G2</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">G2</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">f2</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="Olive">G2</font> &amp; <font color="Maroon">f2</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="Olive">G2</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:28_1_1_4_6_4_3"/>
not <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_4_6_4_3"/>
<font color="Maroon">f</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>, , , <a class="txt" href="group_7.html#E5"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E32:28_1_1_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">b</font> = the <a href="group_1.html#U1">mult</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> <a href="realset1.html#K1">||</a> <font color="Maroon">A</font> as   <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #);<br/>
<a NAME="E11:28_1_1"/>
 <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #) is <a href="group_1.html#V3">Group-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #) <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #);<br/>

<a NAME="E2:28_1_1_5"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM1">Element</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/>;<br/>
<a NAME="E4:28_1_1_5"/><i><font color="Green">E201</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">e</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">e</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> = 
<font color="Maroon">h1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>

<b>thus </b><font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <span class="p2">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>

<b>reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">h1</font>, <font color="Maroon">hh</font> = <font color="Maroon">h1</font> <a href="group_1.html#K3">"</a>  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/>
<a NAME="E8:28_1_1_5"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">h1</font> <a href="group_1.html#K3">"</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">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>, <font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E2:28_1_1_5_3"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:28_1_1_5_3"/><i><font color="Green">E205</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/> 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="Maroon">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="Maroon">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> )
 <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">G</font> being  <a href="group_1.html#NM1">Group</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="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="group_1.html#K3">"</a>  );<br/>
<a NAME="E4:28_1_1_5_3"/><i><font color="Green">E207</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> holds <br/> ex <font color="Olive">j</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_5_3_1"><b>proof </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="E1:28_1_1_5_3_1"/><i><font color="Green">E208</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Gg</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="E3:28_1_1_5_3_1"/><i><font color="Green">E210</font></i>: 
<font color="Maroon">Gg</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/>;<br/>
<a NAME="E4:28_1_1_5_3_1"/>
 ex <font color="Olive">Ga</font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a>  <a href="group_1.html#L1">HGrStr</a>  st <font color="Olive">Ga</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/>
<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">Gg</font> as   <a href="group_1.html#NM1">Group</a> <b>by </b><i/>;<br/>
<a NAME="E6:28_1_1_5_3_1"/>
 ex <font color="Olive">GG</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">GG</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">GG</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Olive">GG</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">m</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">m</font> <a href="group_1.html#K3">"</a> 
;<br/>

<b>thus </b><a NAME="E8:28_1_1_5_3_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">i</font>,<font color="Maroon">m</font> <a href="group_1.html#K3">"</a> ]
 <b>by </b><i/>;<br/>


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

<b>consider </b><font color="Maroon">f'</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><br/><a NAME="E6:28_1_1_5_3"/><i><font color="Green">E213</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">S<sub>2</sub></font>[<font color="Olive">j</font>,<font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3">PBOOLE:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f'</font>
;<br/>

<a NAME="E7:28_1_1_5_3"/><i><font color="Green">E214</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">hh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">gf'</font> = <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font>;<br/>
<a NAME="E8:28_1_1_5_3"/><i><font color="Green">E216</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">J</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f'</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/>
<i><font color="Green">E217</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">f'</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">f'</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="28_1_1_5_3_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="E1:28_1_1_5_3_3"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">G3</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="E3:28_1_1_5_3_3"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">G3</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/>;<br/><a NAME="E4:28_1_1_5_3_3"/>
 ex <font color="Olive">G4</font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V4">associative</a>  <a href="group_1.html#L1">HGrStr</a>  st <font color="Olive">G4</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><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, </i>;<br/><b>then consider </b><font color="Maroon">G5</font> being   <a href="group_1.html#NM1">Group</a><b> such that </b><br/><a NAME="E6:28_1_1_5_3_3"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">G5</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/>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_5_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_5_3_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="28_1_1_5_3_3_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_5_3_3_1_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><a NAME="E2:28_1_1_5_3_3_1_1"/><b>then </b><i><font color="Green">E225</font></i>: 
 ex <font color="Olive">G</font> being  <a href="group_1.html#NM1">Group</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">i</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">i</font> &amp; <font color="Maroon">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="group_1.html#K3">"</a>  )
 <b>by </b><i/>;<br/><a NAME="E3:28_1_1_5_3_3_1_1"/><i><font color="Green">E226</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><a NAME="E4:28_1_1_5_3_3_1_1"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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>, , , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_5_3_3_1_1"/>
<font color="Maroon">hh</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_5_3_3_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_5_3_3_1_2"/><i><font color="Green">E227</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>then </b><i><font color="Green">E228</font></i>: <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, , </i>
;<br/><i><font color="Green">E229</font></i>: <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G3</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, , </i>
;<br/><a NAME="E4:28_1_1_5_3_3_1_2"/>
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G5</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G5</font></span>)</span> <a href="group_1.html#K3">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T16">GROUP_1:16</a></i>;<br/><b>hence </b><a NAME="E5:28_1_1_5_3_3_1_2"/>
<font color="Maroon">hh</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a>, , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:28_1_1_5_3"/>
<font color="Maroon">h1</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f'</font>
 <b>by </b><i>, <a class="ref" href="group_7.html#T1" target="_self">Th1</a>, <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="E12:28_1_1_5_3"/><i><font color="Green">E230</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   <a href="group_1.html#NM1">Group</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="E14:28_1_1_5_3"/><i><font color="Green">E231</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">f'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="group_1.html#K3">"</a>  )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E15:28_1_1_5_3"/>
( <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">f'</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> )
 <b>by </b><i/>;<br/>

<a NAME="E16:28_1_1_5_3"/>
 ex <font color="Olive">G1</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">G1</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">f</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="Olive">G1</font> &amp; <font color="Maroon">f</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="Olive">G1</font> )
 
<b>by </b><i>, <a class="txt" href="group_7.html#E2:28_1_1"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:28_1_1_5_3"/>
<font color="Maroon">f'</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>, <a class="ref" href="group_1.html#T18">GROUP_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:28_1_1_5"/>
 <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> <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>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h1</font> <a href="group_1.html#K3">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #) <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">h'</font>
;<br/>

<a NAME="E11:28_1_1_5"/><i><font color="Green">E233</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h</font>,<font color="Maroon">h'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">h'</font>,<font color="Maroon">h</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">A</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h'</font> = 
<font color="Maroon">h1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/>

<b>thus </b><font color="Maroon">h'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">h1</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> =  <a href="group_1.html#G1">HGrStr</a>(# <font color="Maroon">A</font>,<font color="Maroon">b</font> #) as  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>  ;<br/>
<a NAME="E13:28_1_1"/>
<font color="Maroon">H</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:28_1_1_6"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <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/>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D5">GROUP_2:def 5</a><br/>

<b>thus </b><a NAME="E2:28_1_1_6"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</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> <a href="realset1.html#K1">||</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of  <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font> ;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:28_1_1_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 ;<br/><a NAME="E2:28_1_1_7"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:28_1_1_7"/>
 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> ) ) )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">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>, <font color="Maroon">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>, <font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font><b> such that </b><a NAME="E16:28_1_1"/><i><font color="Green">E234</font></i>: 
( <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> &amp; <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">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="Maroon">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="Maroon">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="Maroon">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> ) ) )
 ;<br/>

<b>set </b><font color="Maroon">gf</font> = <font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font>;<br/>
<a NAME="E17:28_1_1"/><i><font color="Green">E235</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/>
<a NAME="E18:28_1_1"/><i><font color="Green">E236</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</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/>
<i><font color="Green">E237</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">f</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">f</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="28_1_1_9"><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="E1:28_1_1_9"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:28_1_1_9"/><b>then </b><i><font color="Green">E239</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</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="28_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1_9_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="28_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_9_1_1"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><div class="add"><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> <b> such that </b><br/><a NAME="E3:28_1_1_9_1_1"/><i><font color="Green">E241</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> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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>thus </b><a NAME="E4:28_1_1_9_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_9_1_2"/><i><font color="Green">E242</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="E3:28_1_1_9_1_2"/><i><font color="Green">E243</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/><span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">g</font> <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/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, , , </i>
;<br/><b>hence </b><a NAME="E5:28_1_1_9_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="funct_4.html#K1">+*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E21:28_1_1"/><b>then </b>
<font color="Maroon">x</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">F</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E22:28_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div>
