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

<b>set </b><font color="Maroon" title="c3">G</font> =  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c2">F</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</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>  ex <font color="Olive" title="b2">e</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> 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="c1">I</font> &amp; <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font> &amp; ( for <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> holds <br/> ( <font color="Olive" title="b3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> &amp; <font color="Olive" title="b2">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h</font> &amp;  ex <font color="Olive" title="b4">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font> &amp; <font color="Olive" title="b4">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e</font> ) ) ) );<br/>
<a NAME="E1:15_1_1"/><i><font color="Green" title="E10">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/>
<a NAME="E2:15_1_1"/><i><font color="Green" title="E11">A2</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c4">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="c5">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="c6">i1</font> = <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">I1</font> <b>by </b><i><a class="txt" href="group_7.html#E1:15_1_1_1"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">F1</font> = <font color="Maroon" title="c2">F</font> as  <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a> <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c5">I1</font> ;<br/>
<a NAME="E5:15_1_1_1"/>
<font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font> is <a href="group_1.html#V3" title="GROUP_1:attr.3">Group-like</a>
 
<b>by </b><i><a class="ref" href="group_7.html#D6" target="_self" title="GROUP_7:def.6">Def6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font></span>)</span><b> such that </b><br/><a NAME="E7:15_1_1_1"/><i><font color="Green" title="E13">A4</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c8">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font></span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> ) )
 <b>by </b><i><a class="ref" href="group_1.html#D3" title="GROUP_1:def.3">GROUP_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">e</font>
;<br/>

<a NAME="E8:15_1_1_1"/><i><font color="Green" title="E14">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="c6">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="c7">F1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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="E9:15_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="c6">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#E1:15_1_1"><i><font color="Green" title="E10">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="E10:15_1_1_1"/>
<font color="Maroon" title="c8">e</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#E8:15_1_1_1"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font>
;<br/>

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

<b>thus </b><a NAME="E11:15_1_1_1"/>
( <font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</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="c4">i</font> &amp; <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c9">h</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font></span>)</span>;<br/>

<b>thus </b><a NAME="E12:15_1_1_1"/>
( <font color="Maroon" title="c9">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp; <font color="Maroon" title="c8">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">h</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">F1</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c6">i1</font></span>)</span> st <br/>( <font color="Maroon" title="c9">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> &amp; <font color="Olive" title="b1">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e</font> ) )
 <b>by </b><i><a class="txt" href="group_7.html#E7:15_1_1_1"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">ee</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:15_1_1"/><i><font color="Green" title="E12">A6</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">ee</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E5:15_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">ee</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>
 <b>and </b><br/><a NAME="E6:15_1_1"/><i><font color="Green" title="E13">A7</font></i>: 
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>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">ee</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#E2:15_1_1"><i><font color="Green" title="E11">A2</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_2"/><i><font color="Green" title="E14">A8</font></i>: 
<font color="Maroon" title="c5">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> <font color="Maroon" title="c4">ee</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">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="c7">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">Fi</font><b> such that </b><br/><a NAME="E3:15_1_1_2"/><i><font color="Green" title="E15">A9</font></i>: 
( <font color="Maroon" title="c6">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="c5">i</font> &amp; <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">Fi</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c7">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">Fi</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e</font> ) ) ) )
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="group_7.html#E6:15_1_1"><i><font color="Green" title="E13">A7</font></i></a></i>;<br/><a NAME="E4:15_1_1_2"/>
 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="c5">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="c5">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#E4:15_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_2"><i><font color="Green" title="E14">A8</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:15_1_1_2"/>
<font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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="c5">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:15_1_1_2"><i><font color="Green" title="E15">A9</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:15_1_1"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c4">ee</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#E1:15_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">e1</font> = <font color="Maroon" title="c4">ee</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>take </b>
<font color="Maroon" title="c5">e1</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="c6">h</font> be   <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>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c6">h</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>reconsider </b><font color="Maroon" title="c8">he</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c5">e1</font>, <font color="Maroon" title="c9">eh</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">h</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="E12:15_1_1"/><i><font color="Green" title="E15">A11</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c8">he</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c9">eh</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">h1</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#E1:15_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_3"/><i><font color="Green" title="E16">A12</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:15_1_1_3"/><b>then </b><i><font color="Green" title="E17">A13</font></i>: 
 ex <font color="Olive" title="b1">Gi</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">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">Gi</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="c10">i</font> &amp; <font color="Olive" title="b2">f</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="c7">h1</font>,<font color="Maroon" title="c5">e1</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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">Gi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E8:15_1_1"><i><font color="Green" title="E14">A10</font></i></a>, <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="c11">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="c12">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font><b> such that </b><br/><a NAME="E4:15_1_1_3"/><i><font color="Green" title="E18">A14</font></i>: 
( <font color="Maroon" title="c11">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="c10">i</font> &amp; <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c12">e2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> &amp; <font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> ) ) ) )
 <b>by </b><i><a class="txt" href="group_7.html#E6:15_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_3"><i><font color="Green" title="E16">A12</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">h2</font> = <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> <b>by </b><i><a class="txt" href="group_7.html#E1:15_1_1_3"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_3"><i><font color="Green" title="E18">A14</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="E6:15_1_1_3"/>
<font color="Maroon" title="c13">h2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h2</font>
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1_3"><i><font color="Green" title="E18">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E7:15_1_1_3"/>
<font color="Maroon" title="c8">he</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:15_1_1_3"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_3"><i><font color="Green" title="E18">A14</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:15_1_1"/>
<font color="Maroon" title="c6">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:15_1_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_4"/><i><font color="Green" title="E16">A15</font></i>: 
<font color="Maroon" title="c10">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:15_1_1_4"/><b>then </b><i><font color="Green" title="E17">A16</font></i>: 
 ex <font color="Olive" title="b1">Gi</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">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">Gi</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="c10">i</font> &amp; <font color="Olive" title="b2">f</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="c5">e1</font>,<font color="Maroon" title="c7">h1</font> &amp; <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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">Gi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E8:15_1_1"><i><font color="Green" title="E14">A10</font></i></a>, <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="c11">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="c12">e2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font><b> such that </b><br/><a NAME="E4:15_1_1_4"/><i><font color="Green" title="E18">A17</font></i>: 
( <font color="Maroon" title="c11">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="c10">i</font> &amp; <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> &amp; ( for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c12">e2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> &amp; <font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> ) ) ) )
 <b>by </b><i><a class="txt" href="group_7.html#E6:15_1_1"><i><font color="Green" title="E13">A7</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_4"><i><font color="Green" title="E16">A15</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">h2</font> = <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> <b>by </b><i><a class="txt" href="group_7.html#E1:15_1_1_4"><i><font color="Green" title="E16">A15</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_4"><i><font color="Green" title="E18">A17</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="E6:15_1_1_4"/>
<font color="Maroon" title="c12">e2</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">h2</font>
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1_4"><i><font color="Green" title="E18">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E7:15_1_1_4"/>
<font color="Maroon" title="c9">eh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:15_1_1_4"><i><font color="Green" title="E17">A16</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_4"><i><font color="Green" title="E18">A17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:15_1_1"/>
<font color="Maroon" title="c5">e1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font>
 <b>by </b><i><a class="txt" href="group_7.html#E12:15_1_1"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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">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">hi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> 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="c1">I</font> &amp; <font color="Olive" title="b2">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font> &amp;  ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> st <br/>( <font color="Olive" title="b2">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Olive" title="b3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) );<br/>
<a NAME="E17:15_1_1"/><i><font color="Green" title="E16">A18</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</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">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:15_1_1_5"/><i><font color="Green" title="E17">A19</font></i>: 
<font color="Maroon" title="c10">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="c11">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="c12">e</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font><b> such that </b><br/><a NAME="E3:15_1_1_5"/><i><font color="Green" title="E18">A20</font></i>: 
( <font color="Maroon" title="c11">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="c10">i</font> &amp; <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font> )
 <b>and </b><br/><a NAME="E4:15_1_1_5"/><i><font color="Green" title="E19">A21</font></i>: 
for <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> holds <br/> ( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp; <font color="Maroon" title="c12">e</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> &amp;  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font> &amp; <font color="Olive" title="b2">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font> ) )
 <b>by </b><i><a class="txt" href="group_7.html#E6:15_1_1"><i><font color="Green" title="E13">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">hi</font> = <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font> <b>by </b><i><a class="txt" href="group_7.html#E1:15_1_1_5"><i><font color="Green" title="E17">A19</font></i></a>, <a class="txt" href="group_7.html#E3:15_1_1_5"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="group_7.html#E5"><i><font color="Green" title="E2">Lm1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fi</font><b> such that </b><br/><a NAME="E7:15_1_1_5"/><i><font color="Green" title="E20">A22</font></i>: 
( <font color="Maroon" title="c13">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font> &amp; <font color="Maroon" title="c14">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1_5"><i><font color="Green" title="E19">A21</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">g</font>
;<br/>

<a NAME="E8:15_1_1_5"/><i><font color="Green" title="E21">A23</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="c10">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="c10">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#E1:15_1_1_5"><i><font color="Green" title="E17">A19</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E9:15_1_1_5"/>
<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="c10">i</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#E1:15_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_5"><i><font color="Green" title="E17">A19</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="E10:15_1_1_5"/>
<font color="Maroon" title="c14">g</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#E3:15_1_1_5"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="group_7.html#E8:15_1_1_5"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c11">Fi</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c13">hi</font>
;<br/>

<b>thus </b><a NAME="E11:15_1_1_5"/>
( <font color="Maroon" title="c11">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="c10">i</font> &amp; <font color="Maroon" title="c13">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E3:15_1_1_5"><i><font color="Green" title="E18">A20</font></i></a></i>;<br/>

<b>take </b>
<font color="Maroon" title="c14">g</font>
;<br/>

<b>thus </b><a NAME="E12:15_1_1_5"/>
( <font color="Maroon" title="c13">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c14">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E3:15_1_1_5"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="group_7.html#E7:15_1_1_5"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E19:15_1_1"/><i><font color="Green" title="E17">A24</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E20:15_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">gg</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>
 <b>and </b><br/><a NAME="E21:15_1_1"/><i><font color="Green" title="E18">A25</font></i>: 
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="c10">gg</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#E17:15_1_1"><i><font color="Green" title="E16">A18</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_6"/><i><font color="Green" title="E19">A26</font></i>: 
<font color="Maroon" title="c11">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> <font color="Maroon" title="c10">gg</font>
 ;<br/><a NAME="E2:15_1_1_6"/><b>then </b><i><font color="Green" title="E20">A27</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="c11">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="c11">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#E19:15_1_1"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">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="c13">hi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">Fi</font><b> such that </b><br/><a NAME="E4:15_1_1_6"/><i><font color="Green" title="E21">A28</font></i>: 
( <font color="Maroon" title="c12">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="c11">i</font> &amp; <font color="Maroon" title="c13">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">Fi</font> st <br/>( <font color="Maroon" title="c13">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Olive" title="b1">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> &amp; <font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )
 <b>by </b><i><a class="txt" href="group_7.html#E19:15_1_1"><i><font color="Green" title="E17">A24</font></i></a>, <a class="txt" href="group_7.html#E21:15_1_1"><i><font color="Green" title="E18">A25</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_6"><i><font color="Green" title="E19">A26</font></i></a></i>;<br/><b>thus </b><a NAME="E5:15_1_1_6"/>
<font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:15_1_1_6"><i><font color="Green" title="E20">A27</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_6"><i><font color="Green" title="E21">A28</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E23:15_1_1"/><b>then </b><i><font color="Green" title="E19">A29</font></i>: 
<font color="Maroon" title="c10">gg</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#E1:15_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E19:15_1_1"><i><font color="Green" title="E17">A24</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">g1</font> = <font color="Maroon" title="c10">gg</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>take </b>
<font color="Maroon" title="c11">g1</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c12">he</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c6">h</font>,<font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c13">eh</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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c6">h</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="E26:15_1_1"/><i><font color="Green" title="E20">A30</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">he</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">eh</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#E1:15_1_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_7"/><i><font color="Green" title="E21">A31</font></i>: 
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/><a NAME="E2:15_1_1_7"/><b>then </b><i><font color="Green" title="E22">A32</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>  ex <font color="Olive" title="b2">hi</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> 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="c14">i</font> &amp; <font color="Olive" title="b2">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  ex <font color="Olive" title="b3">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">Fi</font> st <br/>( <font color="Olive" title="b2">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b3">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )
 <b>by </b><i><a class="txt" href="group_7.html#E21:15_1_1"><i><font color="Green" title="E18">A25</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">Gi</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="c16">h5</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:15_1_1_7"/><i><font color="Green" title="E23">A33</font></i>: 
( <font color="Maroon" title="c15">Gi</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="c14">i</font> &amp; <font color="Maroon" title="c16">h5</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="c7">h1</font>,<font color="Maroon" title="c10">gg</font> &amp; <font color="Maroon" title="c16">h5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c15">Gi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E23:15_1_1"><i><font color="Green" title="E19">A29</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_7"><i><font color="Green" title="E21">A31</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E5:15_1_1_7"/>
<font color="Maroon" title="c12">he</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E2:15_1_1_7"><i><font color="Green" title="E22">A32</font></i></a>, <a class="txt" href="group_7.html#E4:15_1_1_7"><i><font color="Green" title="E23">A33</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E28:15_1_1"/>
<font color="Maroon" title="c6">h</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="group_7.html#E26:15_1_1"><i><font color="Green" title="E20">A30</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:15_1_1_8"/><i><font color="Green" title="E21">A34</font></i>: 
<font color="Maroon" title="c14">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="c15">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="c16">hi</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">Fi</font><b> such that </b><br/><a NAME="E3:15_1_1_8"/><i><font color="Green" title="E22">A35</font></i>: 
( <font color="Maroon" title="c15">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="c14">i</font> &amp; <font color="Maroon" title="c16">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp;  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">Fi</font> st <br/>( <font color="Maroon" title="c16">hi</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Olive" title="b1">g</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c16">hi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) )
 <b>by </b><i><a class="txt" href="group_7.html#E21:15_1_1"><i><font color="Green" title="E18">A25</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c17">Gi</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="c18">h5</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:15_1_1_8"/><i><font color="Green" title="E23">A36</font></i>: 
( <font color="Maroon" title="c17">Gi</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="c14">i</font> &amp; <font color="Maroon" title="c18">h5</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="c10">gg</font>,<font color="Maroon" title="c7">h1</font> &amp; <font color="Maroon" title="c18">h5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">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="c17">Gi</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">h1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font></span>)</span> )
 <b>by </b><i><a class="txt" href="group_7.html#E23:15_1_1"><i><font color="Green" title="E19">A29</font></i></a>, <a class="txt" href="group_7.html#E1:15_1_1_8"><i><font color="Green" title="E21">A34</font></i></a>, <a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></i>;<br/><b>thus </b><a NAME="E6:15_1_1_8"/>
<font color="Maroon" title="c13">eh</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ee</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:15_1_1_8"><i><font color="Green" title="E22">A35</font></i></a>, <a class="txt" href="group_7.html#E5:15_1_1_8"><i><font color="Green" title="E23">A36</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:15_1_1"/>
<font color="Maroon" title="c11">g1</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="txt" href="group_7.html#E4:15_1_1"><i><font color="Green" title="E12">A6</font></i></a>, <a class="txt" href="group_7.html#E26:15_1_1"><i><font color="Green" title="E20">A30</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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