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<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="group_7.html#NM1" title="GROUP_7:NM.1">HGrStr-Family</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c4">G</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green" title="E10">A1</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> )
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green" title="E11">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <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="c3">F</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><br/>

<b>set </b><font color="Maroon" title="c5">GP</font> =  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F</font>;<br/>
<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a><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>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) ) );<br/>
<a NAME="E3:19"/><i><font color="Green" title="E12">A3</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_1_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c9">j</font>,<font color="Olive" title="b1">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_2"/><i><font color="Green" title="E14">A5</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><a NAME="E2:19_1_1_2"/>
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_1"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:19_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</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> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">Fj</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self" title="GROUP_7:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c10">Fj</font>;<br/><b>take </b>
<font color="Maroon" title="c11">a</font>
;<br/><b>thus </b><a NAME="E6:19_1_1_2"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_1_1_2"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/><b>thus </b><a NAME="E7:19_1_1_2"/>
( <font color="Maroon" title="c9">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">gx</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green" title="E13">A6</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c9">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="group_7.html#E3:19"><i><font color="Green" title="E12">A3</font></i></a>);<br/></i>
<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>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) ) );<br/>
<a NAME="E6:19"/><i><font color="Green" title="E14">A7</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_2"/><i><font color="Green" title="E15">A8</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_2_1"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_1"><b>suppose </b></a><a NAME="E1:19_2_1_1"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_2_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">j</font>,<font color="Olive" title="b1">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_2_1_2"><b>suppose </b></a><a NAME="E1:19_2_1_2"/><i><font color="Green" title="E16">A9</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><a NAME="E2:19_2_1_2"/>
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_2"><i><font color="Green" title="E15">A8</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:19_2_1_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</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> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">Fj</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self" title="GROUP_7:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c11">Fj</font>;<br/><b>take </b>
<font color="Maroon" title="c12">a</font>
;<br/><b>thus </b><a NAME="E6:19_2_1_2"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_2_1_2"><i><font color="Green" title="E16">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E7:19_2_1_2"/>
( <font color="Maroon" title="c10">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> &amp; <font color="Maroon" title="c12">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">gy</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E8:19"/><i><font color="Green" title="E15">A10</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c10">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="group_7.html#E6:19"><i><font color="Green" title="E14">A7</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> ) &amp; ( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) ) );<br/>
<a NAME="E9:19"/><i><font color="Green" title="E16">A11</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/> ex <font color="Olive" title="b2">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Olive" title="b2">k</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_3_1"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> or <font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1"><b>suppose </b></a><a NAME="E1:19_3_1_1"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:19_3_1_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c11">j</font>,<font color="Olive" title="b1">k</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_2"><b>suppose </b></a><a NAME="E1:19_3_1_2"/><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><a NAME="E2:19_3_1_2"/>
<font color="Maroon" title="c11">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_3"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E3:19_3_1_2"/><b>then </b>
<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</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> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">Fj</font> = <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  <b>by </b><i><a class="ref" href="group_7.html#D1" target="_self" title="GROUP_7:def.1">Def1</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">Fj</font>;<br/><b>take </b>
<font color="Maroon" title="c13">a</font>
;<br/><b>thus </b><a NAME="E6:19_3_1_2"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">i</font> implies <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E1:19_3_1_2"><i><font color="Green" title="E18">A13</font></i></a></i>;<br/><b>thus </b><a NAME="E7:19_3_1_2"/>
( <font color="Maroon" title="c11">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> implies  ex <font color="Olive" title="b1">H</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  ex <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">H</font> st <br/>( <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">j</font> &amp; <font color="Maroon" title="c13">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c11">gz</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E11:19"/><i><font color="Green" title="E17">A14</font></i>: 
for <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">j</font>,<font color="Maroon" title="c11">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="group_7.html#E9:19"><i><font color="Green" title="E16">A11</font></i></a>);<br/></i>
<a NAME="E12:19"/><i><font color="Green" title="E18">A15</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">gx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">gy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">gz</font> <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="E13:19"/><i><font color="Green" title="E19">A16</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">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="c3">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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4"/><i><font color="Green" title="E20">A17</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">gx</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E3:19_4"/><i><font color="Green" title="E21">A18</font></i>: 
( <font color="Maroon" title="c13">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="Maroon" title="c13">R</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_4_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font> or <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_1"><b>suppose </b></a><a NAME="E1:19_4_1_1"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j</font>
 ;<br/><div class="add"><a NAME="E2:19_4_1_1"/><b>then </b>
<font color="Maroon" title="c9">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="group_7.html#E5:19"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_4"><i><font color="Green" title="E20">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_4_1_1"/>
<font color="Maroon" title="c9">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E3:19_4"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="group_7.html#E1:19_4_1_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_4_1_2"><b>suppose </b></a><a NAME="E1:19_4_1_2"/>
<font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">H</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> , <font color="Maroon" title="c15">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c14">H</font><b> such that </b><br/><a NAME="E3:19_4_1_2"/><i><font color="Green" title="E22">A20</font></i>: 
( <font color="Maroon" title="c14">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> &amp; <font color="Maroon" title="c9">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E5:19"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_4"><i><font color="Green" title="E20">A17</font></i></a></i>;<br/><b>thus </b><a NAME="E4:19_4_1_2"/>
<font color="Maroon" title="c9">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:19_4"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="group_7.html#E3:19_4_1_2"><i><font color="Green" title="E22">A20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c12">gx</font> = <font color="Maroon" title="c9">gx</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="c3">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E13:19"><i><font color="Green" title="E19">A16</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="19_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_5"/><i><font color="Green" title="E20">A21</font></i>: 
<font color="Maroon" title="c13">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">gy</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E3:19_5"/><i><font color="Green" title="E21">A22</font></i>: 
( <font color="Maroon" title="c14">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</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="Maroon" title="c14">R</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_5_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font> or <font color="Maroon" title="c13">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_5_1_1"><b>suppose </b></a><a NAME="E1:19_5_1_1"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j</font>
 ;<br/><div class="add"><a NAME="E2:19_5_1_1"/><b>then </b>
<font color="Maroon" title="c10">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="group_7.html#E8:19"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_5"><i><font color="Green" title="E20">A21</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_5_1_1"/>
<font color="Maroon" title="c10">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E3:19_5"><i><font color="Green" title="E21">A22</font></i></a>, <a class="txt" href="group_7.html#E1:19_5_1_1"><i><font color="Green" title="E22">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_5_1_2"><b>suppose </b></a><a NAME="E1:19_5_1_2"/>
<font color="Maroon" title="c13">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c15">H</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> , <font color="Maroon" title="c16">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c15">H</font><b> such that </b><br/><a NAME="E3:19_5_1_2"/><i><font color="Green" title="E22">A24</font></i>: 
( <font color="Maroon" title="c15">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> &amp; <font color="Maroon" title="c10">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E8:19"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_5"><i><font color="Green" title="E20">A21</font></i></a></i>;<br/><b>thus </b><a NAME="E4:19_5_1_2"/>
<font color="Maroon" title="c10">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:19_5"><i><font color="Green" title="E21">A22</font></i></a>, <a class="txt" href="group_7.html#E3:19_5_1_2"><i><font color="Green" title="E22">A24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">gy</font> = <font color="Maroon" title="c10">gy</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="c3">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E13:19"><i><font color="Green" title="E19">A16</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="19_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">j</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:19_6"/><i><font color="Green" title="E20">A25</font></i>: 
<font color="Maroon" title="c14">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">gz</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E3:19_6"/><i><font color="Green" title="E21">A26</font></i>: 
( <font color="Maroon" title="c15">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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="Maroon" title="c15">R</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_6_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font> or <font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_1"><b>suppose </b></a><a NAME="E1:19_6_1_1"/><i><font color="Green" title="E22">A27</font></i>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j</font>
 ;<br/><div class="add"><a NAME="E2:19_6_1_1"/><b>then </b>
<font color="Maroon" title="c11">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="group_7.html#E11:19"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_6"><i><font color="Green" title="E20">A25</font></i></a></i>;<br/><b>hence </b><a NAME="E3:19_6_1_1"/>
<font color="Maroon" title="c11">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E3:19_6"><i><font color="Green" title="E21">A26</font></i></a>, <a class="txt" href="group_7.html#E1:19_6_1_1"><i><font color="Green" title="E22">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_6_1_2"><b>suppose </b></a><a NAME="E1:19_6_1_2"/>
<font color="Maroon" title="c14">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">i</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c16">H</font> being  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> , <font color="Maroon" title="c17">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c16">H</font><b> such that </b><br/><a NAME="E3:19_6_1_2"/><i><font color="Green" title="E22">A28</font></i>: 
( <font color="Maroon" title="c16">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> &amp; <font color="Maroon" title="c11">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> )
 <b>by </b><i><a class="txt" href="group_7.html#E11:19"><i><font color="Green" title="E17">A14</font></i></a>, <a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E1:19_6"><i><font color="Green" title="E20">A25</font></i></a></i>;<br/><b>thus </b><a NAME="E4:19_6_1_2"/>
<font color="Maroon" title="c11">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</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="c3">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j</font>
 <b>by </b><i><a class="txt" href="group_7.html#E3:19_6"><i><font color="Green" title="E21">A26</font></i></a>, <a class="txt" href="group_7.html#E3:19_6_1_2"><i><font color="Green" title="E22">A28</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c14">gz</font> = <font color="Maroon" title="c11">gz</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="c3">F</font></span>)</span> <b>by </b><i><a class="txt" href="group_7.html#E12:19"><i><font color="Green" title="E18">A15</font></i></a>, <a class="txt" href="group_7.html#E13:19"><i><font color="Green" title="E19">A16</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x1</font> = <font color="Maroon" title="c12">gx</font>, <font color="Maroon" title="c16">y1</font> = <font color="Maroon" title="c13">gy</font>, <font color="Maroon" title="c17">z1</font> = <font color="Maroon" title="c14">gz</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="c3">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="c18">xy</font> = <font color="Maroon" title="c15">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y1</font>, <font color="Maroon" title="c19">xyz3</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">z1</font>, <font color="Maroon" title="c20">yz</font> = <font color="Maroon" title="c16">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">z1</font>, <font color="Maroon" title="c21">xyz5</font> = <font color="Maroon" title="c15">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">z1</font></span>)</span> as    <a href="pralg_2.html#M1" title="PRALG_2:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c3">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="c22">xyi</font> = <font color="Maroon" title="c18">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c23">yzi</font> = <font color="Maroon" title="c20">yz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">G</font> <b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</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="E23:19"/><i><font color="Green" title="E20">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c16">y1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c17">z1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="group_7.html#E2:19"><i><font color="Green" title="E11">A2</font></i></a></i>;<br/>
<a NAME="E24:19"/><i><font color="Green" title="E21">A30</font></i>: 
( <font color="Maroon" title="c12">gx</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c13">gy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c14">gz</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z</font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E5:19"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="group_7.html#E8:19"><i><font color="Green" title="E15">A10</font></i></a>, <a class="txt" href="group_7.html#E11:19"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/>
<a NAME="E25:19"/><b>then </b>
( <font color="Maroon" title="c18">xy</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c22">xyi</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">xyz3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c23">yzi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">xyz5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> )
 
<b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E26:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_7.html#E1:19"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="group_7.html#E23:19"><i><font color="Green" title="E20">A29</font></i></a>, <a class="txt" href="group_7.html#E24:19"><i><font color="Green" title="E21">A30</font></i></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</a></i>;<br/>


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