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

<b>set </b><font color="Maroon" title="c3">A</font> =  <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/>
<a NAME="E1:9_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
 <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="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>,   <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>,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>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="c1">I</font> holds <br/> ex <font color="Olive" title="b2">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>  ex <font color="Olive" title="b3">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">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="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 &amp; <font color="Olive" title="b3">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b2">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p1">(<span class="default">$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> );<br/>
<a NAME="E2:9_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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="b3">z</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> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</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>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">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>  st <br/>( <font color="Olive" title="b1">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> $1 &amp; $2 <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="Olive" title="b1">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> );<br/>
<a NAME="E1:9_1_1_1"/><i><font color="Green" title="E5">A3</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="c1">I</font> holds <br/> ex <font color="Olive" title="b2">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<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></span>)</span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">w</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:9_1_1_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">I1</font> = <font color="Maroon" title="c1">I</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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c6">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E1:9_1_1_1_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">F1</font> = <font color="Maroon" title="c2">F</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c7">I1</font> ;<br/>
<b>take </b><font color="Maroon" title="c10">w</font> = the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span>;<br/>

<a NAME="E5:9_1_1_1_1"/><i><font color="Green" title="E7">A5</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="c8">i1</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="c9">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</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/>
<a NAME="E6:9_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</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"><font color="Maroon" title="c9">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</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"><font color="Maroon" title="c9">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E7:9_1_1_1_1"/><b>then </b><i><font color="Green" title="E8">A6</font></i>: 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c9">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span> <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"><font color="Maroon" title="c9">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">i1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binop_1.html#T29" title="BINOP_1:th.29">BINOP_1:29</a></i>;<br/>
<a NAME="E8:9_1_1_1_1"/>
<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="c8">i1</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"><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#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:9_1_1_1_1"/>
<font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="txt" href="group_7.html#E5:9_1_1_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="txt" href="group_7.html#E7:9_1_1_1_1"><i><font color="Green" title="E8">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>

<b>thus </b><a NAME="E10:9_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c10">w</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c6">z</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:9_1_1_1"/><i><font color="Green" title="E6">A7</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<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></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="group_7.html#E1:9_1_1_1"><i><font color="Green" title="E5">A3</font></i></a>);<br/></i>
<a NAME="E4:9_1_1_1"/><i><font color="Green" title="E7">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">z</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>
 
<b>by </b><i><a class="txt" href="group_7.html#E3:9_1_1_1"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:9_1_1_1"/>
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>  <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> holds <br/><font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:9_1_1_1_2"/><i><font color="Green" title="E8">A9</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ;<br/>

<a NAME="E2:9_1_1_1_2"/><b>then </b><i><font color="Green" title="E9">A10</font></i>: 
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Olive" title="b1">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="c7">i</font> &amp; <font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="Olive" title="b1">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="group_7.html#E3:9_1_1_1"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/>
<a NAME="E3:9_1_1_1_2"/><i><font color="Green" title="E10">A11</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="c7">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="c7">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="txt" href="group_7.html#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="group_7.html#E1:9_1_1_1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">I1</font> = <font color="Maroon" title="c1">I</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#E1:9_1_1_1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">i1</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E1:9_1_1_1_2"><i><font color="Green" title="E8">A9</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">F1</font> = <font color="Maroon" title="c2">F</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c8">I1</font> ;<br/>
<a NAME="E7:9_1_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i1</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"><font color="Maroon" title="c10">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c9">i1</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i1</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"><font color="Maroon" title="c10">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c9">i1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:9_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="group_7.html#E3:9_1_1_1_2"><i><font color="Green" title="E10">A11</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="E8:9_1_1_1_2"/>
<font color="Maroon" title="c6">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="c7">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:9_1_1_1_2"><i><font color="Green" title="E9">A10</font></i></a>, <a class="txt" href="group_7.html#E3:9_1_1_1_2"><i><font color="Green" title="E10">A11</font></i></a>, <a class="ref" href="binop_1.html#T29" title="BINOP_1:th.29">BINOP_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c6">z</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#E4:9_1_1_1"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">z</font>
;<br/>

<b>let </b><font color="Maroon" title="c8">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E7:9_1_1_1"/>
<font color="Maroon" title="c8">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="c9">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> <b> such that </b><br/><a NAME="E9:9_1_1_1"/><i><font color="Green" title="E8">A12</font></i>: 
( <font color="Maroon" title="c9">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="c8">i</font> &amp; <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c9">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E3:9_1_1_1"><i><font color="Green" title="E6">A7</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">Fi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c7">z</font>
;<br/>

<b>thus </b><a NAME="E10:9_1_1_1"/>
( <font color="Maroon" title="c9">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="c8">i</font> &amp; <font color="Maroon" title="c7">z</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Maroon" title="c7">z</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">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="c9">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E9:9_1_1_1"><i><font color="Green" title="E8">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">B</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</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> such that </b><br/><a NAME="E4:9_1_1"/><i><font color="Green" title="E5">A13</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</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> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c4">B</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="group_7.html#E2:9_1_1"><i><font color="Green" title="E4">A2</font></i></a>);<br/></i>
<b>take </b>
 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<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></span>)</span>,<font color="Maroon" title="c4">B</font> #)
;<br/>

<b>thus </b><a NAME="E5:9_1_1"/>
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>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<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></span>)</span>,<font color="Maroon" title="c4">B</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 ;<br/>

<b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</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>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<b>assume </b><a NAME="E6:9_1_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<b>hence </b><a NAME="E7:9_1_1"/>
 ex <font color="Olive" title="b1">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>  ex <font color="Olive" title="b2">h</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">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="c7">i</font> &amp; <font color="Olive" title="b2">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 <a href="group_1.html#G1" title="GROUP_1:aggr.1">HGrStr</a>(# <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<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></span>)</span>,<font color="Maroon" title="c4">B</font> #) <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> &amp; <font color="Olive" title="b2">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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="Olive" title="b1">Fi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E4:9_1_1"><i><font color="Green" title="E5">A13</font></i></a></i>;<br/>


</div>
