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

<b>set </b><font color="Maroon">G</font> =  <a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Fi</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">e</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Olive">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> ) ) ) );<br/>
<a NAME="E1:18_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:18_1_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">I1</font> = <font color="Maroon">I</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>reconsider </b><font color="Maroon">i1</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I1</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">F1</font> = <font color="Maroon">F</font> as  <a href="group_7.html#V2" target="_self">Group-like</a> <a href="group_7.html#NM1">HGrStr-Family</a> of <font color="Maroon">I1</font> ;<br/>
<a NAME="E5:18_1_1_1"/>
<font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font> is <a href="group_1.html#V3">Group-like</a>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font></span>)</span><b> such that </b><br/><a NAME="E7:18_1_1_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">e</font>
;<br/>

<a NAME="E8:18_1_1_1"/><i><font color="Green">E54</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 
<b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E9:18_1_1_1"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:18_1_1_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>

<b>take </b>
<font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font>
;<br/>

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

<b>thus </b><a NAME="E11:18_1_1_1"/>
( <font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font></span>)</span>;<br/>

<b>thus </b><a NAME="E12:18_1_1_1"/>
( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="group_7.html#K1" target="_self">.</a> <font color="Maroon">i1</font></span>)</span> st <br/>( <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">ee</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:18_1_1"/><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">ee</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E5:18_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ee</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E6:18_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">ee</font>
 ;<br/><b>then consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E3:18_1_1_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) ) ) )
 <b>by </b><i>, </i>;<br/><a NAME="E4:18_1_1_2"/>
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 <b>by </b><i>, , <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:18_1_1_2"/>
<font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E8:18_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">ee</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">ee</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e1</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D3">GROUP_1:def 3</a><br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">he</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">h</font>,<font color="Maroon">e1</font>, <font color="Maroon">eh</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">e1</font>,<font color="Maroon">h</font> as    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E12:18_1_1"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">he</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">eh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:18_1_1_3"/><b>then </b><i><font color="Green">E63</font></i>: 
 ex <font color="Olive">Gi</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h1</font>,<font color="Maroon">e1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Olive">Gi</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E4:18_1_1_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e2</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) ) ) )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> <b>by </b><i>, , </i>;<br/><a NAME="E6:18_1_1_3"/>
<font color="Maroon">h2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:18_1_1_3"/>
<font color="Maroon">he</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E14:18_1_1"/><b>then </b>
<font color="Maroon">he</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E15:18_1_1"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:18_1_1_4"/><b>then </b><i><font color="Green">E66</font></i>: 
 ex <font color="Olive">Gi</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">f</font> being  <a href="funct_1.html#NM1">Function</a> st <br/>( <font color="Olive">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">e1</font>,<font color="Maroon">h1</font> &amp; <font color="Olive">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Olive">Gi</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">e2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E4:18_1_1_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; ( for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e2</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> ) ) ) )
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">h2</font> = <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> <b>by </b><i>, , </i>;<br/><a NAME="E6:18_1_1_4"/>
<font color="Maroon">e2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:18_1_1_4"/>
<font color="Maroon">eh</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E17:18_1_1"/><b>then </b>
<font color="Maroon">eh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E18:18_1_1"/>
<font color="Maroon">e1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Fi</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">hi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">hi</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) );<br/>
<a NAME="E19:18_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">i</font>,<font color="Olive">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:18_1_1_5"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">e</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E3:18_1_1_5"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>and </b><br/><a NAME="E4:18_1_1_5"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> holds <br/> ( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">e</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Olive">h</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">hi</font> = <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E7:18_1_1_5"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">hi</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g</font>
;<br/>

<a NAME="E8:18_1_1_5"/><i><font color="Green">E95</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 
<b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E9:18_1_1_5"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:18_1_1_5"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>

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

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

<b>thus </b><a NAME="E11:18_1_1_5"/>
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i/>;<br/>

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

<b>thus </b><a NAME="E12:18_1_1_5"/>
( <font color="Maroon">hi</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E21:18_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E22:18_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">gg</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E23:18_1_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">x</font>,<font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_6"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font>
 ;<br/><a NAME="E2:18_1_1_6"/><b>then </b><i><font color="Green">E100</font></i>: 
 ex <font color="Olive">R</font> being   <a href="struct_0.html#L1">1-sorted</a>  st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">R</font> )
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">hi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E4:18_1_1_6"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Maroon">hi</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E5:18_1_1_6"/>
<font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E25:18_1_1"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">gg</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">gg</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">g1</font>
;<br/>

<b>reconsider </b><font color="Maroon">he</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">h</font>,<font color="Maroon">g1</font>, <font color="Maroon">eh</font> = the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">g1</font>,<font color="Maroon">h</font> as    <a href="pralg_2.html#M1">Element</a> of  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E28:18_1_1"/><i><font color="Green">E104</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">he</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">eh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_7"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:18_1_1_7"/><b>then </b><i><font color="Green">E106</font></i>: 
 ex <font color="Olive">Fi</font> being non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a>  ex <font color="Olive">hi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">Fi</font> st <br/>( <font color="Olive">hi</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Olive">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Gi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">h5</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:18_1_1_7"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">h1</font>,<font color="Maroon">gg</font> &amp; <font color="Maroon">h5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Gi</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E5:18_1_1_7"/>
<font color="Maroon">he</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E30:18_1_1"/><b>then </b>
<font color="Maroon">he</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E31:18_1_1"/>
<font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1_1_8"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">Fi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">hi</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font><b> such that </b><br/><a NAME="E3:18_1_1_8"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp;  ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Fi</font> st <br/>( <font color="Maroon">hi</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">hi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">g</font> ) )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Gi</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> , <font color="Maroon">h5</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:18_1_1_8"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">Gi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">h5</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" target="_self">product</a> <font color="Maroon">F</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">gg</font>,<font color="Maroon">h1</font> &amp; <font color="Maroon">h5</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Gi</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E6:18_1_1_8"/>
<font color="Maroon">eh</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E33:18_1_1"/><b>then </b>
<font color="Maroon">eh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ee</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E34:18_1_1"/>
<font color="Maroon">g1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e1</font>
 ;<br/>


</div>
