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

<b>set </b><font color="Maroon" title="c3">GP</font> =  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>set </b><font color="Maroon" title="c4">CF</font> =  <a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E2:25_1_1"/><i><font color="Green" title="E14">A1</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<div><i><font color="Green" title="E15">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">p</font> be    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/><b>thus </b> <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">p</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">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" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">J</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> ex <font color="Olive" title="b2">f</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b1">J</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b2">f</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">J</font> holds <br/> ex <font color="Olive" title="b4">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b4">G</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b4">G</font> ) ) );<br/>
<b>consider </b><font color="Maroon" title="c6">A</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:25_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:25_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:25_1_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">A</font>
 ;<br/>

<a NAME="E2:25_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_2"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:25_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">g</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c7">J</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">I</font>;<br/>
<a NAME="E1:25_1_1_3"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">I</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>take </b>
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">I</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c8">f</font>
;<br/>

<b>thus </b><a NAME="E3:25_1_1_3"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">f</font>
 <b>by </b><i><a class="ref" href="funct_4.html#T22" title="FUNCT_4:th.22">FUNCT_4:22</a></i>;<br/>

<b>thus </b><a NAME="E4:25_1_1_3"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> )
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <font color="Maroon" title="c6">A</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font>;<br/>
<a NAME="E9:25_1_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">A</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:25_1_1_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:25_1_1_4"/><b>then </b>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T119" title="ZFMISC_1:th.119">ZFMISC_1:119</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1_4"/><i><font color="Green" title="E19">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon" title="c9">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:25_1_1_4"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:25_1_1_4"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span> &amp; <span class="p1">(<span class="default">the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:25_1_1_4"/><i><font color="Green" title="E21">A8</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_4"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="group_7.html#E6:25_1_1_4"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">J1</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c14">f1</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c13">J1</font><b> such that </b><br/><a NAME="E10:25_1_1_4"/><i><font color="Green" title="E22">A9</font></i>: 
<font color="Maroon" title="c11">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c14">f1</font>
 <b>and </b><br/><a NAME="E11:25_1_1_4"/><i><font color="Green" title="E23">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_4"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">J2</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c16">f2</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c15">J2</font><b> such that </b><br/><a NAME="E13:25_1_1_4"/><i><font color="Green" title="E24">A11</font></i>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f2</font>
 <b>and </b><br/><a NAME="E14:25_1_1_4"/><i><font color="Green" title="E25">A12</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_4"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/>
<a NAME="E15:25_1_1_4"/><i><font color="Green" title="E26">A13</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">J1</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">J2</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">x1</font> = <font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c18">x2</font> = <font color="Maroon" title="c12">x2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a></i>;<br/>
<i><font color="Green" title="E27">A14</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">x1</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">f1</font></span>)</span>
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_4_2"/><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:25_1_1_4_2"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_4_2_1"/>
( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> or  not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_2_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_4_2_1_1"/><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_2_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E11:25_1_1_4"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_4_2_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_2_1_1"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_2_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_4_2_1_2"/><i><font color="Green" title="E30">A18</font></i>: 
not <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c20">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_4_2_1_2"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c20">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/><font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_2_1_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c20">G</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_2_1_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E5:25_1_1_4_2_1_2"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1_4_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_2_1_2"><i><font color="Green" title="E31">A19</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c17">x1</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E17:25_1_1_4"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<i><font color="Green" title="E28">A20</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">x2</font> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">f2</font></span>)</span>
<b>by </b><i><a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_4_4"/><i><font color="Green" title="E29">A21</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:25_1_1_4_4"/><b>then </b><i><font color="Green" title="E30">A22</font></i>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_4_4_1"/>
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> or  not <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_4_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_4_4_1_1"/><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_4_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E14:25_1_1_4"><i><font color="Green" title="E25">A12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_4_4_1_1"/>
<font color="Maroon" title="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_4"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_4_1_1"><i><font color="Green" title="E31">A23</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_4_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_4_4_1_2"/><i><font color="Green" title="E31">A24</font></i>: 
not <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c21">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_4_4_1_2"/><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_4"><i><font color="Green" title="E29">A21</font></i></a>, <a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/><font color="Maroon" title="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_4_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c21">G</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_4"><i><font color="Green" title="E29">A21</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_4_1_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E5:25_1_1_4_4_1_2"/>
<font color="Maroon" title="c18">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1_4_4"><i><font color="Green" title="E30">A22</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_4_1_2"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c20">x2</font> = <font color="Maroon" title="c18">x2</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E20:25_1_1_4"><i><font color="Green" title="E28">A20</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">X1</font> = <font color="Maroon" title="c19">x1</font>, <font color="Maroon" title="c22">X2</font> = <font color="Maroon" title="c20">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c23">ff</font> = <font color="Maroon" title="c21">X1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">X2</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b2">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G</font> );<br/>
<b>consider </b><font color="Maroon" title="c24">K</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E26:25_1_1_4"/><i><font color="Green" title="E29">A26</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">K</font> iff ( <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<div><i><font color="Green" title="E30">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_5"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c25">J</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>take </b><font color="Maroon" title="c26">J</font> = <font color="Maroon" title="c25">J</font>;<br/><b>thus </b><a NAME="E2:25_1_1_4_5"/>
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c23">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c23">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E28:25_1_1_4"/><i><font color="Green" title="E31">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c20">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="group_7.html#E8:25_1_1_4"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:25_1_1_4"/><b>then </b><i><font color="Green" title="E32">A29</font></i>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">X1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c22">X2</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1_4"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_4"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c25">y1</font> = <font color="Maroon" title="c9">y</font> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E31:25_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_6"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c26">J</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c13">J1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">J2</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c24">K</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c26">J</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">k2</font> );<br/>
<a NAME="E2:25_1_1_4_6"/><i><font color="Green" title="E33">A30</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_4_6_1"/><i><font color="Green" title="E34">A31</font></i>: 
<font color="Maroon" title="c27">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_4_6_1"/><i><font color="Green" title="E35">A32</font></i>: 
<font color="Maroon" title="c28">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">j</font>
 <b>by </b><i><a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">k1</font> = <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">j</font>, <font color="Maroon" title="c30">k2</font> = <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c28">G</font> <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_1"><i><font color="Green" title="E34">A31</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_6_1"><i><font color="Green" title="E35">A32</font></i></a>, <a class="txt" href="group_7.html#E5"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c29">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c30">k2</font>
;<br/>

<b>thus </b><a NAME="E5:25_1_1_4_6_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c27">j</font>,<font color="Maroon" title="c29">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c30">k2</font>]
 <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_4_6_1"><i><font color="Green" title="E35">A32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c27">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c26">J</font><b> such that </b><br/><a NAME="E4:25_1_1_4_6"/><i><font color="Green" title="E34">A33</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="group_7.html#E2:25_1_1_4_6"><i><font color="Green" title="E33">A30</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c27">f</font>
;<br/>

<b>set </b><font color="Maroon" title="c28">gf</font> = <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font>;<br/>
<a NAME="E5:25_1_1_4_6"/><i><font color="Green" title="E35">A34</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">J</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E6:25_1_1_4_6"/><i><font color="Green" title="E36">A35</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c25">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<i><font color="Green" title="E37">A36</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c27">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_4_6_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c29">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_4_6_3"/><i><font color="Green" title="E38">A37</font></i>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c30">Fi</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> , <font color="Maroon" title="c31">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:25_1_1_4_6_3"/><i><font color="Green" title="E39">A38</font></i>: 
( <font color="Maroon" title="c30">Fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Maroon" title="c31">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c19">x1</font>,<font color="Maroon" title="c20">x2</font> &amp; <font color="Maroon" title="c31">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c30">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font></span>)</span> )
 <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/><b>consider </b><font color="Maroon" title="c32">FF</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E5:25_1_1_4_6_3"/><i><font color="Green" title="E40">A39</font></i>: 
<font color="Maroon" title="c32">FF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c33">Fi</font> = <font color="Maroon" title="c30">Fi</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_4_6_3"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6_3"><i><font color="Green" title="E40">A39</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c34">x1i</font> = <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>, <font color="Maroon" title="c35">x2i</font> = <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c33">Fi</font> <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_6_3"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="group_7.html#E5"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/><a NAME="E8:25_1_1_4_6_3"/><i><font color="Green" title="E41">A40</font></i>: 
<font color="Maroon" title="c25">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">x1i</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c35">x2i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1_4"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_4"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="group_7.html#E28:25_1_1_4"><i><font color="Green" title="E31">A28</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_6_3"><i><font color="Green" title="E39">A38</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_4_6_3_1"/>
( <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font> or  not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_3_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_4_6_3_1_1"/><i><font color="Green" title="E42">A41</font></i>: 
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_3_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">GG</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ex <font color="Olive" title="b2">k1a</font>, <font color="Olive" title="b3">k2a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">GG</font> st <br/>( <font color="Olive" title="b1">GG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Olive" title="b2">k1a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Olive" title="b3">k2a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k1a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">k2a</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E4:25_1_1_4_6"><i><font color="Green" title="E34">A33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_4_6_3_1_1"/>
<font color="Maroon" title="c25">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E29:25_1_1_4"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6"><i><font color="Green" title="E35">A34</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_6_3"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_3_1_1"><i><font color="Green" title="E42">A41</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_3_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_4_6_3_1_2"/><i><font color="Green" title="E42">A42</font></i>: 
not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E43">A43</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1_4_6"><i><font color="Green" title="E35">A34</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c32">FF</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6_3"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_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="25_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="E1:25_1_1_4_6_3_1_2_2_1"/>
(  not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">J2</font> or <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">K</font> )
 </span><b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_3_1_2"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_3_1_2_2_1_1"><b>case </b></a><a NAME="E1:25_1_1_4_6_3_1_2_2_1_1"/>
not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">J2</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; not <font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:25_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>;<br/><a NAME="E4:25_1_1_4_6_3_1_2_2_1_1"/><b>then </b>
( <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c32">FF</font> &amp; <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c32">FF</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_4_6_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6_3"><i><font color="Green" title="E40">A39</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_4_6_3_1_2_2_1_1"/>
<font color="Maroon" title="c25">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_4_6_3"><i><font color="Green" title="E39">A38</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6_3"><i><font color="Green" title="E40">A39</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_4_6_3"><i><font color="Green" title="E41">A40</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_6_3_1_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_3_1_2_2_1_2"><b>case </b></a><a NAME="E1:25_1_1_4_6_3_1_2_2_1_2"/>
<font color="Maroon" title="c29">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">K</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_3_1_2_2_1_2"/><b>then </b>
 ex <font color="Olive" title="b1">GG</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ex <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">GG</font> st <br/>( <font color="Olive" title="b1">GG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Olive" title="b2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> &amp; <font color="Olive" title="b2">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">GG</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">GG</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">GG</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E26:25_1_1_4"><i><font color="Green" title="E29">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_4_6_3_1_2_2_1_2"/>
<font color="Maroon" title="c25">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E29:25_1_1_4"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_3"><i><font color="Green" title="E38">A37</font></i></a>, <a class="txt" href="group_7.html#E5:25_1_1_4_6_3"><i><font color="Green" title="E40">A39</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_6_3_1_2"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_1_4_6_3_1_2"/>
<font color="Maroon" title="c25">y1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">i</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:25_1_1_4_6"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c27">f</font>
 <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1_4_6"><i><font color="Green" title="E36">A35</font></i></a>, <a class="txt" href="group_7.html#E7:25_1_1_4_6"><i><font color="Green" title="E37">A36</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c29">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E10:25_1_1_4_6"/><i><font color="Green" title="E38">A44</font></i>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c26">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c30">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> , <font color="Maroon" title="c31">k1</font>, <font color="Maroon" title="c32">k2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c30">G</font><b> such that </b><br/><a NAME="E12:25_1_1_4_6"/><i><font color="Green" title="E39">A45</font></i>: 
( <font color="Maroon" title="c30">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c31">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c32">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c31">k1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c32">k2</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E4:25_1_1_4_6"><i><font color="Green" title="E34">A33</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c30">G</font>
;<br/>

<b>thus </b><a NAME="E13:25_1_1_4_6"/>
<font color="Maroon" title="c30">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:25_1_1_4_6"><i><font color="Green" title="E39">A45</font></i></a></i>;<br/>

<b>thus </b><a NAME="E14:25_1_1_4_6"/>
<font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c30">G</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:25_1_1_4_6"><i><font color="Green" title="E39">A45</font></i></a></i>;<br/>

<a NAME="E15:25_1_1_4_6"/><i><font color="Green" title="E40">A46</font></i>: 
<font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c15">J2</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4_6"><i><font color="Green" title="E38">A44</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_4_6_4"/>
( ( <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; not <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> ) or ( not <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> ) or ( <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> ) )
 </span><b>by </b><i><a class="txt" href="group_7.html#E15:25_1_1_4_6"><i><font color="Green" title="E40">A46</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_4_1"><b>suppose </b></a><a NAME="E1:25_1_1_4_6_4_1"/><i><font color="Green" title="E41">A47</font></i>: 
( <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; not <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_4_1"/><b>then </b><i><font color="Green" title="E42">A48</font></i>: 
<font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">G1</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E4:25_1_1_4_6_4_1"/><i><font color="Green" title="E43">A49</font></i>: 
( <font color="Maroon" title="c33">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c33">G1</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c33">G1</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E11:25_1_1_4"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_4_1"><i><font color="Green" title="E41">A47</font></i></a></i>;<br/><font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font>
<b>by </b><i><a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_4_1"><i><font color="Green" title="E41">A47</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c33">G1</font>
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4_6"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="group_7.html#E4:25_1_1_4_6_4_1"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E6:25_1_1_4_6_4_1"/>
<font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c30">G</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:25_1_1_4_6"><i><font color="Green" title="E39">A45</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_6_4_1"><i><font color="Green" title="E42">A48</font></i></a>, <a class="txt" href="group_7.html#E4:25_1_1_4_6_4_1"><i><font color="Green" title="E43">A49</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_4_2"><b>suppose </b></a><a NAME="E1:25_1_1_4_6_4_2"/><i><font color="Green" title="E41">A50</font></i>: 
( not <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_4_2"/><b>then </b><i><font color="Green" title="E42">A51</font></i>: 
<font color="Maroon" title="c20">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E13:25_1_1_4"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>consider </b><font color="Maroon" title="c33">G2</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E4:25_1_1_4_6_4_2"/><i><font color="Green" title="E43">A52</font></i>: 
( <font color="Maroon" title="c33">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c33">G2</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c33">G2</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E14:25_1_1_4"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_4_2"><i><font color="Green" title="E41">A50</font></i></a></i>;<br/><font color="Maroon" title="c19">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font>
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="group_7.html#E15:25_1_1_4"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_4_2"><i><font color="Green" title="E41">A50</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c33">G2</font>
<b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4_6"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="group_7.html#E4:25_1_1_4_6_4_2"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E6:25_1_1_4_6_4_2"/>
<font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c30">G</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:25_1_1_4_6"><i><font color="Green" title="E39">A45</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_6_4_2"><i><font color="Green" title="E42">A51</font></i></a>, <a class="txt" href="group_7.html#E4:25_1_1_4_6_4_2"><i><font color="Green" title="E43">A52</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_4_6_4_3"><b>suppose </b></a><a NAME="E1:25_1_1_4_6_4_3"/><i><font color="Green" title="E41">A53</font></i>: 
( <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">J1</font> &amp; <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J2</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_1_4_6_4_3"/><b>then </b><i><font color="Green" title="E42">A54</font></i>: 
 ex <font color="Olive" title="b1">G1</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c14">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G1</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E11:25_1_1_4"><i><font color="Green" title="E23">A10</font></i></a></i>;<br/><a NAME="E3:25_1_1_4_6_4_3"/><i><font color="Green" title="E43">A55</font></i>: 
 ex <font color="Olive" title="b1">G2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G2</font> &amp; <font color="Maroon" title="c16">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G2</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E14:25_1_1_4"><i><font color="Green" title="E25">A12</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_4_6_4_3"><i><font color="Green" title="E41">A53</font></i></a></i>;<br/><a NAME="E4:25_1_1_4_6_4_3"/>
not <font color="Maroon" title="c29">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">K</font>
 <b>by </b><i><a class="txt" href="group_7.html#E10:25_1_1_4_6"><i><font color="Green" title="E38">A44</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_4_6_4_3"/>
<font color="Maroon" title="c27">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c30">G</font>
 <b>by </b><i><a class="txt" href="group_7.html#E26:25_1_1_4"><i><font color="Green" title="E29">A26</font></i></a>, <a class="txt" href="group_7.html#E10:25_1_1_4_6"><i><font color="Green" title="E38">A44</font></i></a>, <a class="txt" href="group_7.html#E12:25_1_1_4_6"><i><font color="Green" title="E39">A45</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_4_6_4_3"><i><font color="Green" title="E42">A54</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_4_6_4_3"><i><font color="Green" title="E43">A55</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="E32:25_1_1_4"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E27:25_1_1_4"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="group_7.html#E29:25_1_1_4"><i><font color="Green" title="E32">A29</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">b</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> <font color="Maroon" title="c7">A</font> as   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c7">A</font> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">H</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #);<br/>
<a NAME="E11:25_1_1"/>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #) is <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c5">g</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #) <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E7:25_1_1"><i><font color="Green" title="E18">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon" title="c12">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #);<br/>

<a NAME="E2:25_1_1_5"/>
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">h1</font> = <font color="Maroon" title="c12">h</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E4:25_1_1_5"/><i><font color="Green" title="E19">A56</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c11">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c12">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">e</font> = 
<font color="Maroon" title="c13">h1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c12">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><br/>

<b>thus </b><font color="Maroon" title="c11">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">h</font> = 
<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p2">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">h1</font>
<b>by </b><i><a class="txt" href="group_7.html#E4:25_1_1_5"><i><font color="Green" title="E19">A56</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c12">h</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>

<b>reconsider </b><font color="Maroon" title="c14">h2</font> = <font color="Maroon" title="c13">h1</font>, <font color="Maroon" title="c15">hh</font> = <font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<a NAME="E8:25_1_1_5"/><i><font color="Green" title="E20">A57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c16">J</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c17">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c16">J</font><b> such that </b><br/><a NAME="E2:25_1_1_5_3"/><i><font color="Green" title="E21">A58</font></i>: 
<font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c17">f</font>
 <b>and </b><br/><a NAME="E3:25_1_1_5_3"/><i><font color="Green" title="E22">A59</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ex <font color="Olive" title="b2">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  );<br/>
<a NAME="E4:25_1_1_5_3"/><i><font color="Green" title="E23">A60</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font> holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:25_1_1_5_3_1"/><i><font color="Green" title="E24">A61</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">Gg</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_5_3_1"/><i><font color="Green" title="E25">A62</font></i>: 
<font color="Maroon" title="c19">Gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/>
<a NAME="E4:25_1_1_5_3_1"/>
 ex <font color="Olive" title="b1">Ga</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <font color="Olive" title="b1">Ga</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_1"><i><font color="Green" title="E24">A61</font></i></a>, <a class="ref" href="group_7.html#D4" target="_self" title="GROUP_7:def.4">Def4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">G</font> = <font color="Maroon" title="c19">Gg</font> as   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3_1"><i><font color="Green" title="E25">A62</font></i></a></i>;<br/>
<a NAME="E6:25_1_1_5_3_1"/>
 ex <font color="Olive" title="b1">GG</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">GG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">GG</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">GG</font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3"><i><font color="Green" title="E22">A59</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_5_3_1"><i><font color="Green" title="E24">A61</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">m</font> = <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c20">G</font> <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3_1"><i><font color="Green" title="E25">A62</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c21">m</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
;<br/>

<b>thus </b><a NAME="E8:25_1_1_5_3_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c18">i</font>,<font color="Maroon" title="c21">m</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> ]
 <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3_1"><i><font color="Green" title="E25">A62</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon" title="c16">J</font>
;<br/>

<b>consider </b><font color="Maroon" title="c18">f'</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c16">J</font><b> such that </b><br/><a NAME="E6:25_1_1_5_3"/><i><font color="Green" title="E24">A63</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="group_7.html#E4:25_1_1_5_3"><i><font color="Green" title="E23">A60</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c18">f'</font>
;<br/>

<a NAME="E7:25_1_1_5_3"/><i><font color="Green" title="E25">A64</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">hh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c19">gf'</font> = <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font>;<br/>
<a NAME="E8:25_1_1_5_3"/><i><font color="Green" title="E26">A65</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c17">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">J</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">J</font> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E27">A66</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c18">f'</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_5_3"><i><font color="Green" title="E26">A65</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_5_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_5_3_3"/><i><font color="Green" title="E28">A67</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c21">G3</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_5_3_3"/><i><font color="Green" title="E29">A68</font></i>: 
<font color="Maroon" title="c21">G3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/><a NAME="E4:25_1_1_5_3_3"/>
 ex <font color="Olive" title="b1">G4</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <font color="Olive" title="b1">G4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_3"><i><font color="Green" title="E28">A67</font></i></a>, <a class="ref" href="group_7.html#D4" target="_self" title="GROUP_7:def.4">Def4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">G5</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a><b> such that </b><br/><a NAME="E6:25_1_1_5_3_3"/><i><font color="Green" title="E30">A69</font></i>: 
<font color="Maroon" title="c22">G5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3_3"><i><font color="Green" title="E29">A68</font></i></a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_5_3_3_1"/>
( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font> or  not <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_5_3_3_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_5_3_3_1_1"/><i><font color="Green" title="E31">A70</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_5_3_3_1_1"/><b>then </b><i><font color="Green" title="E32">A71</font></i>: 
 ex <font color="Olive" title="b1">G</font> being  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ex <font color="Olive" title="b2">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">m</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  )
 <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1_5_3"><i><font color="Green" title="E24">A63</font></i></a></i>;<br/><a NAME="E3:25_1_1_5_3_3_1_1"/><i><font color="Green" title="E33">A72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E8:25_1_1_5_3"><i><font color="Green" title="E26">A65</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_5_3_3_1_1"><i><font color="Green" title="E31">A70</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><a NAME="E4:25_1_1_5_3_3_1_1"/>
<font color="Maroon" title="c14">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1_5_3"><i><font color="Green" title="E21">A58</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_5_3"><i><font color="Green" title="E26">A65</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_5_3_3_1_1"><i><font color="Green" title="E31">A70</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_5_3_3_1_1"/>
<font color="Maroon" title="c15">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_3"><i><font color="Green" title="E28">A67</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_5_3_3_1_1"><i><font color="Green" title="E32">A71</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_5_3_3_1_1"><i><font color="Green" title="E33">A72</font></i></a>, <a class="ref" href="group_7.html#T11" target="_self" title="GROUP_7:th.11">Th11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_5_3_3_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_5_3_3_1_2"/><i><font color="Green" title="E31">A73</font></i>: 
not <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green" title="E32">A74</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E8:25_1_1_5_3"><i><font color="Green" title="E26">A65</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c21">G3</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_3"><i><font color="Green" title="E28">A67</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_5_3_3"><i><font color="Green" title="E29">A68</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><i><font color="Green" title="E33">A75</font></i>: <font color="Maroon" title="c14">h2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1_5_3"><i><font color="Green" title="E21">A58</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_5_3"><i><font color="Green" title="E26">A65</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_5_3_3_1_2"><i><font color="Green" title="E31">A73</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c21">G3</font>
<b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_3"><i><font color="Green" title="E28">A67</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_5_3_3"><i><font color="Green" title="E29">A68</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><a NAME="E4:25_1_1_5_3_3_1_2"/>
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c22">G5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c22">G5</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T16" title="GROUP_1:th.16">GROUP_1:16</a></i>;<br/><b>hence </b><a NAME="E5:25_1_1_5_3_3_1_2"/>
<font color="Maroon" title="c15">hh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_5_3_3"><i><font color="Green" title="E28">A67</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_5_3_3"><i><font color="Green" title="E29">A68</font></i></a>, <a class="txt" href="group_7.html#E6:25_1_1_5_3_3"><i><font color="Green" title="E30">A69</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_5_3_3_1_2"><i><font color="Green" title="E32">A74</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_5_3_3_1_2"><i><font color="Green" title="E33">A75</font></i></a>, <a class="ref" href="group_7.html#T11" target="_self" title="GROUP_7:th.11">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:25_1_1_5_3"/>
<font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c18">f'</font>
 <b>by </b><i><a class="txt" href="group_7.html#E7:25_1_1_5_3"><i><font color="Green" title="E25">A64</font></i></a>, <a class="txt" href="group_7.html#E9:25_1_1_5_3"><i><font color="Green" title="E27">A66</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c20">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E12:25_1_1_5_3"/><i><font color="Green" title="E28">A76</font></i>: 
<font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c21">G</font> being   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>, <font color="Maroon" title="c22">m</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c21">G</font><b> such that </b><br/><a NAME="E14:25_1_1_5_3"/><i><font color="Green" title="E29">A77</font></i>: 
( <font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c22">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">m</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  )
 <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1_5_3"><i><font color="Green" title="E24">A63</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c21">G</font>
;<br/>

<b>thus </b><a NAME="E15:25_1_1_5_3"/>
( <font color="Maroon" title="c21">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c21">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E14:25_1_1_5_3"><i><font color="Green" title="E29">A77</font></i></a></i>;<br/>

<a NAME="E16:25_1_1_5_3"/>
 ex <font color="Olive" title="b1">G1</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G1</font> &amp; <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b1">G1</font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1_5_3"><i><font color="Green" title="E22">A59</font></i></a>, <a class="txt" href="group_7.html#E12:25_1_1_5_3"><i><font color="Green" title="E28">A76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:25_1_1_5_3"/>
<font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c21">G</font>
 <b>by </b><i><a class="txt" href="group_7.html#E14:25_1_1_5_3"><i><font color="Green" title="E29">A77</font></i></a>, <a class="ref" href="group_1.html#T18" title="GROUP_1:th.18">GROUP_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:25_1_1_5"/>
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">h'</font> = <font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #) <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E8:25_1_1_5"><i><font color="Green" title="E20">A57</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c16">h'</font>
;<br/>

<a NAME="E11:25_1_1_5"/><i><font color="Green" title="E21">A78</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">h</font>,<font color="Maroon" title="c16">h'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">h'</font>,<font color="Maroon" title="c12">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c12">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c16">h'</font> = 
<font color="Maroon" title="c13">h1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c11">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
;<br/><br/>

<b>thus </b><font color="Maroon" title="c16">h'</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">h1</font>
<b>by </b><i><a class="txt" href="group_7.html#E11:25_1_1_5"><i><font color="Green" title="E21">A78</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c11">e</font>
<b>by </b><i><a class="ref" href="group_1.html#D6" title="GROUP_1:def.6">GROUP_1:def 6</a></i>
;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">H</font> =  <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c9">b</font> #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  ;<br/>
<a NAME="E13:25_1_1"/>
<font color="Maroon" title="c11">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_6"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:25_1_1_6"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_7.html#E6:25_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D5" title="GROUP_2:def.5">GROUP_2:def 5</a><br/>

<b>thus </b><a NAME="E2:25_1_1_6"/>
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <font color="Maroon" title="c11">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="realset1.html#K1" title="REALSET1:func.1">||</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">H</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">H</font> = <font color="Maroon" title="c11">H</font> as  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">H</font>
;<br/>

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:25_1_1_7"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</font>
 ;<br/><a NAME="E2:25_1_1_7"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">x</font>]
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a></i>;<br/><b>hence </b><a NAME="E3:25_1_1_7"/>
 ex <font color="Olive" title="b1">g</font> being   <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> ex <font color="Olive" title="b2">J</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> ex <font color="Olive" title="b3">f</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Olive" title="b2">J</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Olive" title="b3">f</font> &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> holds <br/> ex <font color="Olive" title="b5">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b5">G</font> &amp; <font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b5">G</font> ) ) )
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c14">g</font> being    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>, <font color="Maroon" title="c15">J</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c16">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c15">J</font><b> such that </b><a NAME="E17:25_1_1"/><i><font color="Green" title="E19">A79</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font></span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font> holds <br/> ex <font color="Olive" title="b2">G</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> &amp; <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">G</font> &amp; <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Olive" title="b2">G</font> ) ) )
 ;<br/>

<b>set </b><font color="Maroon" title="c17">gf</font> = <font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font>;<br/>
<a NAME="E18:25_1_1"/><i><font color="Green" title="E20">A80</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="txt" href="group_7.html#E3:25_1_1"><i><font color="Green" title="E15">A2</font></i></a></i>;<br/>
<a NAME="E19:25_1_1"/><i><font color="Green" title="E21">A81</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">J</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<i><font color="Green" title="E22">A82</font></i>:  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#D1" title="FUNCT_4:def.1">FUNCT_4:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="txt" href="group_7.html#E18:25_1_1"><i><font color="Green" title="E20">A80</font></i></a>, <a class="txt" href="group_7.html#E19:25_1_1"><i><font color="Green" title="E21">A81</font></i></a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c18">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1_1_9"/><i><font color="Green" title="E23">A83</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:25_1_1_9"/><b>then </b><i><font color="Green" title="E24">A84</font></i>: 
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> )
 <b>by </b><i><a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_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="E1:25_1_1_9_1"/>
( <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font> or  not <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_9_1_1"/><i><font color="Green" title="E25">A85</font></i>: 
<font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c19">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_9_1_1"/><i><font color="Green" title="E26">A86</font></i>: 
( <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> &amp; <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c19">G</font> &amp; <font color="Maroon" title="c16">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c19">G</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E17:25_1_1"><i><font color="Green" title="E19">A79</font></i></a></i>;<br/><b>thus </b><a NAME="E4:25_1_1_9_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E19:25_1_1"><i><font color="Green" title="E21">A81</font></i></a>, <a class="txt" href="group_7.html#E2:25_1_1_9"><i><font color="Green" title="E24">A84</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_9_1_1"><i><font color="Green" title="E25">A85</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_9_1_1"><i><font color="Green" title="E26">A86</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_9_1_2"/><i><font color="Green" title="E25">A87</font></i>: 
not <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">J</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c19">G</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>  <a href="group_1.html#L1" title="GROUP_1:struct.1">HGrStr</a> <b> such that </b><br/><a NAME="E3:25_1_1_9_1_2"/><i><font color="Green" title="E26">A88</font></i>: 
<font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:25_1_1_9"><i><font color="Green" title="E23">A83</font></i></a>, <a class="ref" href="group_7.html#D3" target="_self" title="GROUP_7:def.3">Def3</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> = 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
<b>by </b><i><a class="txt" href="group_7.html#E19:25_1_1"><i><font color="Green" title="E21">A81</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_9_1_2"><i><font color="Green" title="E25">A87</font></i></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
 <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c19">G</font>
<b>by </b><i><a class="txt" href="group_7.html#E17:25_1_1"><i><font color="Green" title="E19">A79</font></i></a>, <a class="txt" href="group_7.html#E1:25_1_1_9"><i><font color="Green" title="E23">A83</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_9_1_2"><i><font color="Green" title="E26">A88</font></i></a>, <a class="ref" href="group_7.html#T9" target="_self" title="GROUP_7:th.9">Th9</a></i>
;<br/><b>hence </b><a NAME="E5:25_1_1_9_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c16">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1_9"><i><font color="Green" title="E24">A84</font></i></a>, <a class="txt" href="group_7.html#E3:25_1_1_9_1_2"><i><font color="Green" title="E26">A88</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E22:25_1_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_7.html#E2:25_1_1"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="group_7.html#E17:25_1_1"><i><font color="Green" title="E19">A79</font></i></a>, <a class="txt" href="group_7.html#E20:25_1_1"><i><font color="Green" title="E22">A82</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E23:25_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</font>
 <b>by </b><i><a class="txt" href="group_7.html#E5:25_1_1"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="group_7.html#E17:25_1_1"><i><font color="Green" title="E19">A79</font></i></a></i>;<br/>


</div>
