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

<b>let </b><font color="Maroon" title="c1">I</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b2">A</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b3">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">I</font> holds   <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b2">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Olive" title="b3">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">I</font> holds   <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Olive" title="b1">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Olive" title="b2">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EqR</font> being  <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">I</font> holds   <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Olive" title="b1">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/><b>let </b><font color="Maroon" title="c4">EqR</font> be   <a href="eqrel_1.html#NM2" title="EQREL_1:NM.2">Equivalence_Relation</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> </span><br/>







<b>set </b><font color="Maroon" title="c5">U1</font> =  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>;<br/>
<b>set </b><font color="Maroon" title="c6">U2</font> =  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">U1'</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">U2'</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span>;<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> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">g1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b3">e</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> holds  <font color="Olive" title="b2">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">e</font>;<br/>
<a NAME="E1:40"/><i><font color="Green" title="E31">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/> ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> implies 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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">s</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:40_1"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c9">s</font>] )</span><br/>

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">s</font>] ) )</span><br/>

<b>assume </b><a NAME="E2:40_1"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">s</font>] )</span><br/>

<b>reconsider </b><font color="Maroon" title="c11">s'</font> = <font color="Maroon" title="c9">s</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E1:40_1"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<a NAME="E4:40_1"/><i><font color="Green" title="E34">A4</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E2:40_1"><i><font color="Green" title="E33">A3</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c12">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c13">y</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font><b> such that </b><br/><a NAME="E7:40_1"/><i><font color="Green" title="E35">A5</font></i>: 
for <font color="Olive" title="b1">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> holds <br/><font color="Maroon" title="c13">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">c</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S4" title="PBOOLE:sch.4">PBOOLE:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c13">y</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">s</font>] )</span><br/>

<a NAME="E8:40_1"/>
<font color="Maroon" title="c13">y</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E36">A6</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> = 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">y</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:40_1_1"/><i><font color="Green" title="E37">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:40_1_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c13">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">a</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> implies <font color="Maroon" title="c13">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> )</span><br/>

<b>assume </b><a NAME="E1:40_1_1_2"/><i><font color="Green" title="E38">A8</font></i>: 
<font color="Maroon" title="c14">a</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span><br/>

<b>set </b><font color="Maroon" title="c15">A'</font> =  <a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span>;<br/>
<a NAME="E2:40_1_1_2"/><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c14">a</font>
 
<b>by </b><i><a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="pralg_3.html#D5" target="_self" title="PRALG_3:def.5">Def5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:40_1_1_2"/><i><font color="Green" title="E40">A10</font></i>: 
<font color="Maroon" title="c16">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E5:40_1_1_2"/><i><font color="Green" title="E41">A11</font></i>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c16">b</font>
 <b>by </b><i><a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">a</font> = <font color="Maroon" title="c14">a</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E4:40_1_1_2"><i><font color="Green" title="E40">A10</font></i></a>, <a class="txt" href="#E5:40_1_1_2"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<i><font color="Green" title="E42">A12</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">a</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">a</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">a</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c17">a</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">Aa1</font> = <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">a</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c17">a</font> <b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:40_1_1_2"/>
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="c17">a</font> holds <br/><font color="Maroon" title="c18">Aa1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">a</font> implies <font color="Maroon" title="c18">Aa1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font> )</span><br/>

<b>assume </b><a NAME="E1:40_1_1_2_2"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c19">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">Aa1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font></span><br/>

<a NAME="E2:40_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D12" title="PRALG_2:def.12">PRALG_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_1_2_2"/>
<font color="Maroon" title="c18">Aa1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i</font> is  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>  <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="#E7:40_1_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="#E1:40_1_1_2_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">Aa</font> = <font color="Maroon" title="c18">Aa1</font> as    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="pralg_2.html#D12" title="PRALG_2:def.12">PRALG_2:def 12</a></i>;<br/>
<a NAME="E11:40_1_1_2"/>
<font color="Maroon" title="c12">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">a</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:40_1"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="#E4:40_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="pralg_3.html#T1" target="_self" title="PRALG_3:th.1">Th1</a></i>;<br/>
<a NAME="E12:40_1_1_2"/><b>then </b><i><font color="Green" title="E43">A14</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c19">Aa</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T2" target="_self" title="PRALG_3:th.2">Th2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Ji</font> being  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> , <font color="Maroon" title="c21">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c20">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E14:40_1_1_2"/><i><font color="Green" title="E44">A15</font></i>: 
<font color="Maroon" title="c20">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font>
 <b>and </b><br/><a NAME="E15:40_1_1_2"/><i><font color="Green" title="E45">A16</font></i>: 
<font color="Maroon" title="c21">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font>
 <b>and </b><br/><a NAME="E16:40_1_1_2"/><i><font color="Green" title="E46">A17</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c21">Cs</font>
 <b>by </b><i><a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/>
<a NAME="E17:40_1_1_2"/><i><font color="Green" title="E47">A18</font></i>: 
<font color="Maroon" title="c20">Ji</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="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E14:40_1_1_2"><i><font color="Green" title="E44">A15</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E19:40_1_1_2"/><i><font color="Green" title="E48">A19</font></i>: 
( <font color="Maroon" title="c22">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c22">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> )
 <b>by </b><i><a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E20:40_1_1_2"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c19">Aa</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:40_1_1_2"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="#E15:40_1_1_2"><i><font color="Green" title="E45">A16</font></i></a>, <a class="txt" href="#E16:40_1_1_2"><i><font color="Green" title="E46">A17</font></i></a>, <a class="txt" href="#E17:40_1_1_2"><i><font color="Green" title="E47">A18</font></i></a>, <a class="txt" href="#E19:40_1_1_2"><i><font color="Green" title="E48">A19</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>hence </b><a NAME="E21:40_1_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="#E7:40_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="#E2:40_1_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="#E1:40_1_1_2"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="#E12:40_1_1_2"><i><font color="Green" title="E43">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_1_1"/>
<font color="Maroon" title="c13">y</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_1_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:40_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">s</font>] )
 <b>by </b><i><a class="txt" href="#E7:40_1"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c9">F</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:40"/><i><font color="Green" title="E32">A20</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> holds <br/> ex <font color="Olive" title="b2">f</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">s</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1" title="MSSUBFAM:sch.1">MSSUBFAM:sch 1</a>(<a class="txt" href="#E1:40"><i><font color="Green" title="E31">A1</font></i></a>);<br/></i>
<a NAME="E4:40"/><i><font color="Green" title="E33">A21</font></i>: 
<font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c2">S</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:40_2"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/>

<b>thus </b><a NAME="E2:40_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon" title="c12">s</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font>;<br/>
<b>set </b><font color="Maroon" title="c13">U3</font> =  <a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span><b> such that </b><br/><a NAME="E2:40_2_1"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
 <b>and </b><br/><a NAME="E3:40_2_1"/><i><font color="Green" title="E35">A23</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>,<font color="Olive" title="b1">x'</font>, <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font>]
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/>
<a NAME="E4:40_2_1"/><i><font color="Green" title="E36">A24</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<i><font color="Green" title="E37">A25</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>
;<br/>
<a NAME="E6:40_2_1"/><i><font color="Green" title="E38">A26</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>
 
;<br/>
<a NAME="E7:40_2_1"/><b>then </b><i><font color="Green" title="E39">A27</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/>
<a NAME="E8:40_2_1"/><i><font color="Green" title="E40">A28</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_3.html#T20" target="_self" title="PRALG_3:th.20">Th20</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:40_2_1_2"/>
(  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2_1"><b>suppose </b></a><a NAME="E1:40_2_1_2_1"/><i><font color="Green" title="E41">A29</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_2_1_2_1"/><b>then </b>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T11" title="PRALG_2:th.11">PRALG_2:11</a></i>;<br/><a NAME="E3:40_2_1_2_1"/><b>then </b><i><font color="Green" title="E42">A30</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:40_2_1_2_1"/><b>then </b><i><font color="Green" title="E43">A31</font></i>: 
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:40_2_1"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/><i><font color="Green" title="E44">A32</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:40_2_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="txt" href="#E3:40_2_1_2_1"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E6:40_2_1_2_1"/><i><font color="Green" title="E45">A33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_2_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="#E8:40_2_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="txt" href="#E3:40_2_1_2_1"><i><font color="Green" title="E42">A30</font></i></a>, <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="c15">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E4:40_2_1"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/><i><font color="Green" title="E46">A34</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g1</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:40_2_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="#E6:40_2_1_2_1"><i><font color="Green" title="E45">A33</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><a NAME="E9:40_2_1_2_1"/>
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c10">o</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p0">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="pralg_3.html#T9" target="_self" title="PRALG_3:th.9">Th9</a></i>;<br/><a NAME="E10:40_2_1_2_1"/><b>then </b><i><font color="Green" title="E47">A35</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>;<br/><div><i><font color="Green" title="E48">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">e</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">e</font></span>)</span></span><br/><b>consider </b><font color="Maroon" title="c17">Ji</font> being  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> , <font color="Maroon" title="c18">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c17">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E2:40_2_1_2_1_3"/><i><font color="Green" title="E49">A37</font></i>: 
( <font color="Maroon" title="c17">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">e</font> &amp; <font color="Maroon" title="c18">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font> )
 <b>by </b><i><a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/><i><font color="Green" title="E50">A38</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font></span>)</span> = 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="txt" href="#E5:40_2_1_2_1"><i><font color="Green" title="E44">A32</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E4:40_2_1_2_1_3"/>
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c17">Ji</font>,<span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c10">o</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c18">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Olive" title="b1">i'</font></span>)</span></span>)</span> where <font color="Olive" title="b1">i'</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">Ji</font> : verum </span>}</span> </span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="pralg_3.html#T9" target="_self" title="PRALG_3:th.9">Th9</a></i>;<br/><b>then </b><i><font color="Green" title="E51">A39</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font></span>)</span></span>)</span> = 
<font color="Maroon" title="c17">Ji</font>
<b>by </b><i><a class="ref" href="funct_2.html#T169" title="FUNCT_2:th.169">FUNCT_2:169</a></i>
<br/>.= 
<font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="txt" href="#E2:40_2_1_2_1_3"><i><font color="Green" title="E49">A37</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_1_3_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c19">i1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">e</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_1_3_3"/><i><font color="Green" title="E52">A40</font></i>: 
<font color="Maroon" title="c19">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font></span><br/><b>then reconsider </b><font color="Maroon" title="c20">ii</font> = <font color="Maroon" title="c19">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c17">Ji</font> <b>by </b><i><a class="txt" href="#E2:40_2_1_2_1_3"><i><font color="Green" title="E49">A37</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c21">ii'</font> = <font color="Maroon" title="c20">ii</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E1:40_2_1_2_1_3_3"><i><font color="Green" title="E52">A40</font></i></a></i>;<br/><i><font color="Green" title="E53">A41</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c16">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:40_2_1_2_1_3_3"/>
<font color="Maroon" title="c18">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font>
 <b>by </b><i><a class="txt" href="#E2:40_2_1_2_1_3"><i><font color="Green" title="E49">A37</font></i></a>, <a class="ref" href="pralg_3.html#D5" target="_self" title="PRALG_3:def.5">Def5</a></i>;<br/><a NAME="E6:40_2_1_2_1_3_3"/><b>then </b><i><font color="Green" title="E54">A42</font></i>: 
<font color="Maroon" title="c18">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c20">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">ii'</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_1_3_3"><i><font color="Green" title="E52">A40</font></i></a>, <a class="txt" href="#E4:40_2_1_2_1_3_3"><i><font color="Green" title="E53">A41</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font> = 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font>
<b>by </b><i><a class="txt" href="#E3:40_2_1_2_1_3"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="#E1:40_2_1_2_1_3_3"><i><font color="Green" title="E52">A40</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c21">ii'</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="pralg_3.html#T10" target="_self" title="PRALG_3:th.10">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c18">Cs</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">i1</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E6:40_2_1_2_1_3_3"><i><font color="Green" title="E54">A42</font></i></a>, <a class="ref" href="pralg_3.html#T10" target="_self" title="PRALG_3:th.10">Th10</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E7:40_2_1_2_1_3"/>
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c16">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c16">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:40_2_1_2_1_3"><i><font color="Green" title="E49">A37</font></i></a>, <a class="txt" href="#E3:40_2_1_2_1_3"><i><font color="Green" title="E50">A38</font></i></a>, <a class="txt" href="#E5:40_2_1_2_1_3"><i><font color="Green" title="E51">A39</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c16">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> implies <font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_1_4"/>
<font color="Maroon" title="c16">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font></span><br/><b>then reconsider </b><font color="Maroon" title="c17">e</font> = <font color="Maroon" title="c16">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> ;<br/><b>consider </b><font color="Maroon" title="c18">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:40_2_1_2_1_4"/><i><font color="Green" title="E49">A43</font></i>: 
<font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span>
 ;<br/><a NAME="E5:40_2_1_2_1_4"/>
<font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E2:40_2_1"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="#E3:40_2_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="#E4:40_2_1_2_1"><i><font color="Green" title="E43">A31</font></i></a>, <a class="txt" href="#E4:40_2_1_2_1_4"><i><font color="Green" title="E49">A43</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font> = 
 <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:40_2_1_2_1"><i><font color="Green" title="E48">A36</font></i></a>, <a class="txt" href="#E4:40_2_1_2_1_4"><i><font color="Green" title="E49">A43</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x1</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="pralg_3.html#T10" target="_self" title="PRALG_3:th.10">Th10</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><a NAME="E13:40_2_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_3.html#K2" title="PRALG_3:func.2">const</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E8:40_2_1_2_1"><i><font color="Green" title="E46">A34</font></i></a>, <a class="txt" href="#E10:40_2_1_2_1"><i><font color="Green" title="E47">A35</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/><b>hence </b><a NAME="E14:40_2_1_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_1"><i><font color="Green" title="E41">A29</font></i></a>, <a class="txt" href="#E3:40_2_1_2_1"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="pralg_3.html#T12" target="_self" title="PRALG_3:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_2_1_2_2"><b>suppose </b></a><a NAME="E1:40_2_1_2_2"/><i><font color="Green" title="E41">A44</font></i>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:40_2_1_2_2"/><i><font color="Green" title="E42">A45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_2_1"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="#E8:40_2_1"><i><font color="Green" title="E40">A28</font></i></a>, <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="c15">g1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E4:40_2_1"><i><font color="Green" title="E36">A24</font></i></a></i>;<br/><a NAME="E4:40_2_1_2_2"/><i><font color="Green" title="E43">A46</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_3.html#T20" target="_self" title="PRALG_3:th.20">Th20</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">f1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><i><font color="Green" title="E44">A47</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">g1</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:40_2_1"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="#E2:40_2_1_2_2"><i><font color="Green" title="E42">A45</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><i><font color="Green" title="E45">A48</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">f1</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:40_2_1_2_2"><i><font color="Green" title="E43">A46</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><div><i><font color="Green" title="E46">A49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">e</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f2</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> holds <br/><font color="Olive" title="b1">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c18">f2</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> implies <font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_2_3"/><i><font color="Green" title="E47">A50</font></i>: 
<font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span><br/><b>consider </b><font color="Maroon" title="c19">Ji</font> being  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> , <font color="Maroon" title="c20">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c19">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:40_2_1_2_2_3"/><i><font color="Green" title="E48">A51</font></i>: 
( <font color="Maroon" title="c19">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> &amp; <font color="Maroon" title="c20">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c20">Cs</font> )
 <b>by </b><i><a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">f2</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:40_2_1"><i><font color="Green" title="E40">A28</font></i></a>, <a class="txt" href="#E1:40_2_1_2_2_3"><i><font color="Green" title="E47">A50</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><b>then </b><i><font color="Green" title="E49">A52</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> = 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E6:40_2_1_2_2_3"/><i><font color="Green" title="E50">A53</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="ref" href="pralg_3.html#T21" target="_self" title="PRALG_3:th.21">Th21</a></i>;<br/><a NAME="E7:40_2_1_2_2_3"/><i><font color="Green" title="E51">A54</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c20">Cs</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="pralg_3.html#T21" target="_self" title="PRALG_3:th.21">Th21</a></i>;<br/><a NAME="E8:40_2_1_2_2_3"/><i><font color="Green" title="E52">A55</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span> <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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c20">Cs</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="txt" href="#E6:40_2_1_2_2_3"><i><font color="Green" title="E50">A53</font></i></a>, <a class="ref" href="pralg_3.html#T20" target="_self" title="PRALG_3:th.20">Th20</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c21">f3</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><i><font color="Green" title="E53">A56</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c21">f3</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c20">Cs</font>,<span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:40_2_1_2_2_3"><i><font color="Green" title="E52">A55</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c19">Ji</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/><div><i><font color="Green" title="E54">A57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_2_3_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">i1</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">e</font> implies <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_2_3_4"/><i><font color="Green" title="E55">A58</font></i>: 
<font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span><br/><b>then reconsider </b><font color="Maroon" title="c23">i2</font> = <font color="Maroon" title="c22">i1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Ji</font> <b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><a NAME="E3:40_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c23">i2</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="txt" href="#E6:40_2_1_2_2_3"><i><font color="Green" title="E50">A53</font></i></a>, <a class="ref" href="pralg_3.html#T21" target="_self" title="PRALG_3:th.21">Th21</a></i>;<br/><a NAME="E4:40_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c23">i2</font></span>)</span>
 ;<br/><a NAME="E5:40_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</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">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c20">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c23">i2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/><b>then </b><i><font color="Green" title="E56">A59</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c20">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c23">i2</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>
;<br/><a NAME="E7:40_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="ref" href="pralg_3.html#T21" target="_self" title="PRALG_3:th.21">Th21</a></i>;<br/><a NAME="E8:40_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c10">o</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c22">i1</font></span>)</span>
 ;<br/><a NAME="E9:40_2_1_2_2_3_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</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">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>;<br/><b>then </b><i><font color="Green" title="E57">A60</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
<b>by </b><i><a class="ref" href="pralg_2.html#T10" title="PRALG_2:th.10">PRALG_2:10</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_2_3_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">n</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_2_3_4_3"/><i><font color="Green" title="E58">A61</font></i>: 
<font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font></span><br/><a NAME="E2:40_2_1_2_2_3_4_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span>
 <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="c25">s1</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E4:40_2_1_2_2_3_4_3"/><i><font color="Green" title="E59">A62</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="pralg_3.html#T16" target="_self" title="PRALG_3:th.16">Th16</a></i>;<br/><a NAME="E5:40_2_1_2_2_3_4_3"/><b>then </b><i><font color="Green" title="E60">A63</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">f4</font> = <font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E4:40_2_1_2_2_3_4_3"><i><font color="Green" title="E59">A62</font></i></a></i>;<br/><a NAME="E7:40_2_1_2_2_3_4_3"/><i><font color="Green" title="E61">A64</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">SORTS</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font>
 <b>by </b><i><a class="txt" href="#E5:40_2_1_2_2_3_4_3"><i><font color="Green" title="E60">A63</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/> <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c26">f4</font> = 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c25">s1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E5:40_2_1_2_2_3_4_3"><i><font color="Green" title="E60">A63</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/><b>then </b><i><font color="Green" title="E62">A65</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f4</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> = 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E10:40_2_1_2_2_3_4_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="pralg_3.html#T16" target="_self" title="PRALG_3:th.16">Th16</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">f5</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><b>consider </b><font color="Maroon" title="c28">f7</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font><b> such that </b><br/><a NAME="E13:40_2_1_2_2_3_4_3"/><i><font color="Green" title="E63">A66</font></i>: 
<font color="Maroon" title="c28">f7</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font>
 <b>and </b><br/><a NAME="E14:40_2_1_2_2_3_4_3"/><i><font color="Green" title="E64">A67</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c28">f7</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>,<font color="Olive" title="b1">x'</font>,<font color="Maroon" title="c25">s1</font>]
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E32">A20</font></i></a></i>;<br/><font color="Maroon" title="c27">f5</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c10">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="pralg_3.html#T14" target="_self" title="PRALG_3:th.14">Th14</a></i>
<br/>.= 
<font color="Maroon" title="c28">f7</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="#E13:40_2_1_2_2_3_4_3"><i><font color="Green" title="E63">A66</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><a NAME="E16:40_2_1_2_2_3_4_3"/><b>then </b><i><font color="Green" title="E65">A68</font></i>: 
<font color="Maroon" title="c27">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">f4</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E7:40_2_1_2_2_3_4_3"><i><font color="Green" title="E61">A64</font></i></a>, <a class="txt" href="#E14:40_2_1_2_2_3_4_3"><i><font color="Green" title="E64">A67</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c29">f6</font> = <font color="Maroon" title="c27">f5</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E18:40_2_1_2_2_3_4_3"/>
<font color="Maroon" title="c29">f6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="pralg_3.html#T22" target="_self" title="PRALG_3:th.22">Th22</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">f4</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</font>
<b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="txt" href="#E6:40_2_1_2_2_3"><i><font color="Green" title="E50">A53</font></i></a>, <a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="txt" href="#E16:40_2_1_2_2_3_4_3"><i><font color="Green" title="E65">A68</font></i></a>, <a class="ref" href="pralg_3.html#T22" target="_self" title="PRALG_3:th.22">Th22</a></i>
<br/>.= 
<font color="Maroon" title="c26">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4"><i><font color="Green" title="E55">A58</font></i></a>, <a class="txt" href="#E9:40_2_1_2_2_3_4_3"><i><font color="Green" title="E62">A65</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">n</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_4_3"><i><font color="Green" title="E58">A61</font></i></a>, <a class="ref" href="pralg_3.html#T22" target="_self" title="PRALG_3:th.22">Th22</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E12:40_2_1_2_2_3_4"/>
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p4">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i1</font>
 <b>by </b><i><a class="txt" href="#E6:40_2_1_2_2_3_4"><i><font color="Green" title="E56">A59</font></i></a>, <a class="txt" href="#E10:40_2_1_2_2_3_4"><i><font color="Green" title="E57">A60</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_2_3_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c22">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c22">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">e</font> implies <font color="Maroon" title="c21">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_2_3_5"/><i><font color="Green" title="E55">A69</font></i>: 
<font color="Maroon" title="c22">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font></span><br/><b>then reconsider </b><font color="Maroon" title="c23">i2</font> = <font color="Maroon" title="c22">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Ji</font> <b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">i1</font> = <font color="Maroon" title="c23">i2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_5"><i><font color="Green" title="E55">A69</font></i></a></i>;<br/><i><font color="Green" title="E56">A70</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c17">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/><a NAME="E5:40_2_1_2_2_3_5"/>
<font color="Maroon" title="c20">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font>
 <b>by </b><i><a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="ref" href="pralg_3.html#D5" target="_self" title="PRALG_3:def.5">Def5</a></i>;<br/><a NAME="E6:40_2_1_2_2_3_5"/><b>then </b>
<font color="Maroon" title="c20">Cs</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c24">i1</font>
 <b>by </b><i><a class="txt" href="#E1:40_2_1_2_2_3_5"><i><font color="Green" title="E55">A69</font></i></a>, <a class="txt" href="#E4:40_2_1_2_2_3_5"><i><font color="Green" title="E56">A70</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><font color="Maroon" title="c21">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c24">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p5">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">i2</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="#E3:40_2_1_2_2_3"><i><font color="Green" title="E48">A51</font></i></a>, <a class="txt" href="#E7:40_2_1_2_2_3"><i><font color="Green" title="E51">A54</font></i></a>, <a class="ref" href="pralg_3.html#T23" target="_self" title="PRALG_3:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c24">i1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">i1</font></span>)</span>
<b>by </b><i><a class="txt" href="#E11:40_2_1_2_2_3"><i><font color="Green" title="E54">A57</font></i></a>, <a class="txt" href="#E1:40_2_1_2_2_3_5"><i><font color="Green" title="E55">A69</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c18">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="txt" href="#E1:40_2_1_2_2_3"><i><font color="Green" title="E47">A50</font></i></a>, <a class="ref" href="pralg_3.html#T23" target="_self" title="PRALG_3:th.23">Th23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">x1</font>
<b>by </b><i><a class="txt" href="#E5:40_2_1_2_2_3"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="#E1:40_2_1_2_2_3_5"><i><font color="Green" title="E55">A69</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E13:40_2_1_2_2_3"/>
<font color="Maroon" title="c18">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E5:40_2_1_2_2_3"><i><font color="Green" title="E49">A52</font></i></a>, <a class="txt" href="#E10:40_2_1_2_2_3"><i><font color="Green" title="E53">A56</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1_2_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> implies <font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> )</span><br/><b>assume </b><a NAME="E1:40_2_1_2_2_4"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font></span><br/><b>then reconsider </b><font color="Maroon" title="c18">e</font> = <font color="Maroon" title="c17">x1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> ;<br/><b>reconsider </b><font color="Maroon" title="c19">f2</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E8:40_2_1"><i><font color="Green" title="E40">A28</font></i></a></i>;<br/><a NAME="E4:40_2_1_2_2_4"/>
<font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">e</font>
 <b>by </b><i><a class="txt" href="#E2:40_2_1"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="#E3:40_2_1"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="#E7:40_2_1"><i><font color="Green" title="E39">A27</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c15">g1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font> = 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">e</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <span class="p3">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font></span>)</span>
<b>by </b><i><a class="txt" href="#E8:40_2_1_2_2"><i><font color="Green" title="E46">A49</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c16">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x1</font>
<b>by </b><i><a class="txt" href="#E1:40_2_1_2_2"><i><font color="Green" title="E41">A44</font></i></a>, <a class="ref" href="pralg_3.html#T23" target="_self" title="PRALG_3:th.23">Th23</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E10:40_2_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c10">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p3">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c10">o</font>,<span class="p2">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p3">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c11">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E6:40_2_1_2_2"><i><font color="Green" title="E44">A47</font></i></a>, <a class="txt" href="#E7:40_2_1_2_2"><i><font color="Green" title="E45">A48</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E5:40"/><i><font color="Green" title="E34">A71</font></i>: 
<font color="Maroon" title="c9">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> or  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> )</span><br/>

<b>assume </b><a NAME="E1:40_3"/><i><font color="Green" title="E35">A72</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="relat_1.html#K2" title="RELAT_1:func.2">proj2</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">s'</font> = <font color="Maroon" title="c10">s</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E1:40_3"><i><font color="Green" title="E35">A72</font></i></a></i>;<br/>
<a NAME="E3:40_3"/><i><font color="Green" title="E36">A73</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font><b> such that </b><br/><a NAME="E5:40_3"/><i><font color="Green" title="E37">A75</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E6:40_3"/><i><font color="Green" title="E38">A76</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">s</font>]
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E1:40_3"><i><font color="Green" title="E35">A72</font></i></a></i>;<br/>
<a NAME="E7:40_3"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> iff  ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> iff  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<b>thus </b><a NAME="E1:40_3_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> iff  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_3_1_1"/>
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> implies  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_3_1_1_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<a NAME="E2:40_3_1_1_1"/><b>then </b><i><font color="Green" title="E39">A77</font></i>: 
<font color="Maroon" title="c13">y</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:40_3_1_1_1"/><i><font color="Green" title="E40">A78</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">gg</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:40_3_1_1_1"/><i><font color="Green" title="E41">A79</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c14">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c13">y</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E2:40_3_1_1_1"><i><font color="Green" title="E39">A77</font></i></a></i>;<br/>
<a NAME="E7:40_3_1_1_1"/><i><font color="Green" title="E42">A80</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>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">e</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> ex <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">e</font> &amp; <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E8:40_3_1_1_1"/><i><font color="Green" title="E43">A81</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">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c16">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies  ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Olive" title="b1">j</font>] )</span><br/>

<b>assume </b><a NAME="E1:40_3_1_1_1_1"/><i><font color="Green" title="E44">A82</font></i>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<a NAME="E2:40_3_1_1_1_1"/><i><font color="Green" title="E45">A83</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:40_3_1_1_1_1"/>
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c16">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 
<b>by </b><i><a class="txt" href="#E1:40_3_1_1_1_1"><i><font color="Green" title="E44">A82</font></i></a>, <a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font><b> such that </b><br/><a NAME="E5:40_3_1_1_1_1"/><i><font color="Green" title="E46">A84</font></i>: 
<font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c16">i</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c18">Ji</font> being  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> , <font color="Maroon" title="c19">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c18">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E7:40_3_1_1_1_1"/>
<font color="Maroon" title="c18">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>and </b><br/><a NAME="E8:40_3_1_1_1_1"/>
<font color="Maroon" title="c19">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font>
 <b>and </b><br/><a NAME="E9:40_3_1_1_1_1"/><i><font color="Green" title="E47">A85</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c19">Cs</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E11:40_3_1_1_1_1"/><i><font color="Green" title="E48">A86</font></i>: 
( <font color="Maroon" title="c20">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c17">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c20">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E12:40_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c19">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E9:40_3_1_1_1_1"><i><font color="Green" title="E47">A85</font></i></a>, <a class="txt" href="#E11:40_3_1_1_1_1"><i><font color="Green" title="E48">A86</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c21">y'</font> = <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E4:40_3_1_1_1"><i><font color="Green" title="E40">A78</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1"><i><font color="Green" title="E41">A79</font></i></a>, <a class="txt" href="#E2:40_3_1_1_1_1"><i><font color="Green" title="E45">A83</font></i></a>, <a class="txt" href="pralg_3.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<a NAME="E14:40_3_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Maroon" title="c21">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i</font>]
 
<b>by </b><i><a class="txt" href="#E1:40_3_1_1_1_1"><i><font color="Green" title="E44">A82</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1_1"><i><font color="Green" title="E46">A84</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<b>hence </b><a NAME="E15:40_3_1_1_1_1"/>
 ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">i</font>,<font color="Olive" title="b1">j</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c16">x</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="E10:40_3_1_1_1"/><i><font color="Green" title="E44">A87</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/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="#E8:40_3_1_1_1"><i><font color="Green" title="E43">A81</font></i></a>);<br/></i>
<a NAME="E11:40_3_1_1_1"/><i><font color="Green" title="E45">A88</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_2"><b>proof </b></a><div class="add">

<i><font color="Green" title="E46">A89</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">x</font> = 
<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 href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E2:40_3_1_1_1_2"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c17">z</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> implies <font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/>

<b>assume </b><a NAME="E1:40_3_1_1_1_2_2"/>
<font color="Maroon" title="c17">z</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> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/>

<a NAME="E2:40_3_1_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>, <font color="Maroon" title="c19">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:40_3_1_1_1_2_2"/><i><font color="Green" title="E47">A90</font></i>: 
( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">e</font> &amp; <font color="Maroon" title="c19">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> )
 <b>and </b><br/><a NAME="E5:40_3_1_1_1_2_2"/><i><font color="Green" title="E48">A91</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="#E10:40_3_1_1_1"><i><font color="Green" title="E44">A87</font></i></a></i>;<br/>
<a NAME="E6:40_3_1_1_1_2_2"/><i><font color="Green" title="E49">A92</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="#E4:40_3_1_1_1"><i><font color="Green" title="E40">A78</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1"><i><font color="Green" title="E41">A79</font></i></a>, <a class="txt" href="#E7:40_3_1_1_1"><i><font color="Green" title="E42">A80</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c20">Ji</font> being  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> , <font color="Maroon" title="c21">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c20">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E8:40_3_1_1_1_2_2"/><i><font color="Green" title="E50">A93</font></i>: 
<font color="Maroon" title="c20">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font>
 <b>and </b><br/><a NAME="E9:40_3_1_1_1_2_2"/><i><font color="Green" title="E51">A94</font></i>: 
<font color="Maroon" title="c21">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font>
 <b>and </b><br/><a NAME="E10:40_3_1_1_1_2_2"/><i><font color="Green" title="E52">A95</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c21">Cs</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c22">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E12:40_3_1_1_1_2_2"/><i><font color="Green" title="E53">A96</font></i>: 
( <font color="Maroon" title="c22">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c18">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c22">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E13:40_3_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:40_3_1_1_1_2_2"><i><font color="Green" title="E52">A95</font></i></a>, <a class="txt" href="#E12:40_3_1_1_1_2_2"><i><font color="Green" title="E53">A96</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c23">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E15:40_3_1_1_1_2_2"/><i><font color="Green" title="E54">A97</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c23">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E16:40_3_1_1_1_2_2"/><i><font color="Green" title="E55">A98</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c23">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="txt" href="#E6:40_3_1_1_1_2_2"><i><font color="Green" title="E49">A92</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c18">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c18">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c18">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<a NAME="E18:40_3_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E56">A99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 
<b>by </b><i><a class="txt" href="#E4:40_3_1_1_1_2_2"><i><font color="Green" title="E47">A90</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<a NAME="E19:40_3_1_1_1_2_2"/><i><font color="Green" title="E57">A100</font></i>: 
<font color="Maroon" title="c21">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="#E9:40_3_1_1_1_2_2"><i><font color="Green" title="E51">A94</font></i></a>, <a class="ref" href="pralg_3.html#D5" target="_self" title="PRALG_3:def.5">Def5</a></i>;<br/>
<a NAME="E20:40_3_1_1_1_2_2"/>
<font color="Maroon" title="c20">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="#E8:40_3_1_1_1_2_2"><i><font color="Green" title="E50">A93</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>;<br/>
<a NAME="E21:40_3_1_1_1_2_2"/><b>then </b><i><font color="Green" title="E58">A101</font></i>: 
 <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c18">e</font>
 
<b>by </b><i><a class="txt" href="#E19:40_3_1_1_1_2_2"><i><font color="Green" title="E57">A100</font></i></a>, <a class="ref" href="pralg_3.html#T2" target="_self" title="PRALG_3:th.2">Th2</a></i>;<br/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c21">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> = 
<font color="Maroon" title="c20">Ji</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c18">e</font>
<b>by </b><i><a class="txt" href="#E8:40_3_1_1_1_2_2"><i><font color="Green" title="E50">A93</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<b>hence </b><a NAME="E23:40_3_1_1_1_2_2"/>
<font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
 <b>by </b><i><a class="txt" href="#E4:40_3_1_1_1_2_2"><i><font color="Green" title="E47">A90</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1_2_2"><i><font color="Green" title="E48">A91</font></i></a>, <a class="txt" href="#E15:40_3_1_1_1_2_2"><i><font color="Green" title="E54">A97</font></i></a>, <a class="txt" href="#E16:40_3_1_1_1_2_2"><i><font color="Green" title="E55">A98</font></i></a>, <a class="txt" href="#E18:40_3_1_1_1_2_2"><i><font color="Green" title="E56">A99</font></i></a>, <a class="txt" href="#E21:40_3_1_1_1_2_2"><i><font color="Green" title="E58">A101</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40_3_1_1_1_2"/>
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font>
 <b>by </b><i><a class="txt" href="#E3:40_3"><i><font color="Green" title="E36">A73</font></i></a>, <a class="txt" href="#E1:40_3_1_1_1_2"><i><font color="Green" title="E46">A89</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:40_3_1_1_1"/><b>then </b><i><font color="Green" title="E46">A102</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:40_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">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/>
<a NAME="E14:40_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 
;<br/>
<a NAME="E15:40_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">gg'</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E17:40_3_1_1_1"/><i><font color="Green" title="E47">A103</font></i>: 
( <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">gg'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">gg'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E18:40_3_1_1_1"/>
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c17">gg'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">f'</font> = <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">x</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E17:40_3_1_1_1"><i><font color="Green" title="E47">A103</font></i></a></i>;<br/>
<a NAME="E20:40_3_1_1_1"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:40_3_1_1_1_3"/>
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> holds <br/><font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> implies <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font> )</span><br/>

<b>assume </b><a NAME="E1:40_3_1_1_1_3_1"/>
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c20">e</font> = <font color="Maroon" title="c19">e</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font> ;<br/>
<a NAME="E3:40_3_1_1_1_3_1"/><i><font color="Green" title="E48">A104</font></i>: 
<font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font>
 
<b>by </b><i><a class="txt" href="#E4:40_3_1_1_1"><i><font color="Green" title="E40">A78</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1"><i><font color="Green" title="E41">A79</font></i></a>, <a class="txt" href="#E7:40_3_1_1_1"><i><font color="Green" title="E42">A80</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c21">Ji</font> being  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> , <font color="Maroon" title="c22">Cs</font> being    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c21">Ji</font>,<font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E5:40_3_1_1_1_3_1"/><i><font color="Green" title="E49">A105</font></i>: 
<font color="Maroon" title="c21">Ji</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K1" title="PRALG_3:func.1">id</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font>
 <b>and </b><br/><a NAME="E6:40_3_1_1_1_3_1"/>
<font color="Maroon" title="c22">Cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font>
 <b>and </b><br/><a NAME="E7:40_3_1_1_1_3_1"/><i><font color="Green" title="E50">A106</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c22">Cs</font>
 <b>by </b><i><a class="ref" href="pralg_3.html#D6" target="_self" title="PRALG_3:def.6">Def6</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c23">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E9:40_3_1_1_1_3_1"/><i><font color="Green" title="E51">A107</font></i>: 
( <font color="Maroon" title="c23">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c20">e</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c23">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">s'</font> )
 <b>by </b><i><a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/>
<a NAME="E10:40_3_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c22">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E7:40_3_1_1_1_3_1"><i><font color="Green" title="E50">A106</font></i></a>, <a class="txt" href="#E9:40_3_1_1_1_3_1"><i><font color="Green" title="E51">A107</font></i></a>, <a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:40_3_1_1_1_3_1"/><i><font color="Green" title="E52">A108</font></i>: 
( <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">g</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c24">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c22">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> )
 <b>and </b><br/><a NAME="E13:40_3_1_1_1_3_1"/>
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c22">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> holds <br/><font color="Maroon" title="c24">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c22">Cs</font>,<font color="Maroon" title="c11">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="txt" href="#E3:40_3_1_1_1_3_1"><i><font color="Green" title="E48">A104</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">y'</font> = <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">e</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E12:40_3_1_1_1_3_1"><i><font color="Green" title="E52">A108</font></i></a></i>;<br/>
<a NAME="E15:40_3_1_1_1_3_1"/>
<font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<i><font color="Green" title="E53">A109</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">e</font>
<b>by </b><i><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c1">I</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">e</font>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c20">e</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<i><font color="Green" title="E54">A110</font></i>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c25">y'</font> = 
<font color="Maroon" title="c21">Ji</font>
<b>by </b><i><a class="txt" href="#E12:40_3_1_1_1_3_1"><i><font color="Green" title="E52">A108</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c20">e</font>
<b>by </b><i><a class="txt" href="#E5:40_3_1_1_1_3_1"><i><font color="Green" title="E49">A105</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>
<a NAME="E3:40_3_1_1_1_3_1_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">e</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_1_3_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c26">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">e</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> )</span><br/>

<b>assume </b><a NAME="E1:40_3_1_1_1_3_1_1_3"/><i><font color="Green" title="E55">A111</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">e</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font></span><br/>

<b>then consider </b><font color="Maroon" title="c27">e1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>, <font color="Maroon" title="c28">f1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:40_3_1_1_1_3_1_1_3"/><i><font color="Green" title="E56">A112</font></i>: 
<font color="Maroon" title="c26">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">e1</font>
 <b>and </b><br/><a NAME="E4:40_3_1_1_1_3_1_1_3"/><i><font color="Green" title="E57">A113</font></i>: 
<font color="Maroon" title="c28">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">e1</font>
 <b>and </b><br/><a NAME="E5:40_3_1_1_1_3_1_1_3"/><i><font color="Green" title="E58">A114</font></i>: 
<font color="Maroon" title="c16">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="#E10:40_3_1_1_1"><i><font color="Green" title="E44">A87</font></i></a></i>;<br/>
<a NAME="E6:40_3_1_1_1_3_1_1_3"/>
<font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">e1</font>
 
<b>by </b><i><a class="txt" href="#E1:40_3_1_1_1_3_1_1_3"><i><font color="Green" title="E55">A111</font></i></a>, <a class="txt" href="#E3:40_3_1_1_1_3_1_1_3"><i><font color="Green" title="E56">A112</font></i></a>, <a class="ref" href="pralg_3.html#T3" target="_self" title="PRALG_3:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E7:40_3_1_1_1_3_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">i</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_1_1_3_1_1"><i><font color="Green" title="E53">A109</font></i></a>, <a class="txt" href="#E1:40_3_1_1_1_3_1_1_3"><i><font color="Green" title="E55">A111</font></i></a>, <a class="txt" href="#E4:40_3_1_1_1_3_1_1_3"><i><font color="Green" title="E57">A113</font></i></a>, <a class="txt" href="#E5:40_3_1_1_1_3_1_1_3"><i><font color="Green" title="E58">A114</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:40_3_1_1_1_3_1_1"/>
<font color="Maroon" title="c16">x</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">y'</font>
 <b>by </b><i><a class="txt" href="#E1:40_3_1_1_1_3_1_1"><i><font color="Green" title="E53">A109</font></i></a>, <a class="txt" href="#E2:40_3_1_1_1_3_1_1"><i><font color="Green" title="E54">A110</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:40_3_1_1_1_3_1"/>
<font color="Maroon" title="c15">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">e</font>
 <b>by </b><i><a class="txt" href="#E6:40_3"><i><font color="Green" title="E38">A76</font></i></a>, <a class="txt" href="#E11:40_3_1_1_1"><i><font color="Green" title="E45">A88</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:40_3_1_1_1_3"/>
<font color="Maroon" title="c15">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f'</font>
 <b>by </b><i><a class="txt" href="#E4:40_3_1_1_1"><i><font color="Green" title="E40">A78</font></i></a>, <a class="txt" href="#E7:40_3_1_1_1"><i><font color="Green" title="E42">A80</font></i></a>, <a class="txt" href="#E17:40_3_1_1_1"><i><font color="Green" title="E47">A103</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E21:40_3_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="#E12:40_3_1_1_1"><i><font color="Green" title="E46">A102</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:40_3_1_1"/>
(  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_3_1_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:40_3_1_1_2"/><i><font color="Green" title="E39">A115</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">f</font>
 <b>and </b><br/><a NAME="E3:40_3_1_1_2"/><i><font color="Green" title="E40">A116</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span><br/>

<a NAME="E4:40_3_1_1_2"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 
<b>by </b><i><a class="txt" href="#E2:40_3_1_1_2"><i><font color="Green" title="E39">A115</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="E5:40_3_1_1_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <b>by </b><i><a class="txt" href="#E3:40_3_1_1_2"><i><font color="Green" title="E40">A116</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:40_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <b>by </b><i><a class="txt" href="#E5:40_3"><i><font color="Green" title="E37">A75</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:40"/><i><font color="Green" title="E35">A117</font></i>: 
<font color="Maroon" title="c9">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D2" title="MSUALG_3:def.2">MSUALG_3:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">F</font> or  not <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/><b>let </b><font color="Maroon" title="c11">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">F</font> or  not <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> or <font color="Maroon" title="c11">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>



<b>assume </b><a NAME="E1:40_4"/><i><font color="Green" title="E36">A118</font></i>: 
( <font color="Maroon" title="c10">s</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">F</font> &amp; <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">g</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/>

<b>let </b><font color="Maroon" title="c12">x1</font>, <font color="Maroon" title="c13">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c11">g</font> or  not <font color="Maroon" title="c13">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c11">g</font> or  not <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font> or <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:40_4"/><i><font color="Green" title="E37">A119</font></i>: 
<font color="Maroon" title="c12">x1</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">g</font>
 <b>and </b><br/><a NAME="E3:40_4"/><i><font color="Green" title="E38">A120</font></i>: 
<font color="Maroon" title="c13">x2</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">g</font>
 <b>and </b><br/><a NAME="E4:40_4"/><i><font color="Green" title="E39">A121</font></i>: 
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font></span><br/>

<a NAME="E5:40_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">F</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="c2">S</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font><b> such that </b><br/><a NAME="E7:40_4"/><i><font color="Green" title="E40">A122</font></i>: 
<font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font>
 <b>and </b><br/><a NAME="E8:40_4"/><i><font color="Green" title="E41">A123</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">s</font>]
 <b>by </b><i><a class="txt" href="#E3:40"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="#E1:40_4"><i><font color="Green" title="E36">A118</font></i></a></i>;<br/>
<a NAME="E9:40_4"/><i><font color="Green" title="E42">A124</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">g</font>
 
<b>by </b><i><a class="txt" href="#E1:40_4"><i><font color="Green" title="E36">A118</font></i></a>, <a class="txt" href="#E7:40_4"><i><font color="Green" title="E40">A122</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:40_4"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c15">s'</font> = <font color="Maroon" title="c10">s</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="#E1:40_4"><i><font color="Green" title="E36">A118</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E2:40_4_1"/><i><font color="Green" title="E43">A125</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">s'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<a NAME="E3:40_4_1"/><i><font color="Green" title="E44">A126</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <span class="p2">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>,<font color="Maroon" title="c15">s'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pralg_2.html#D17" title="PRALG_2:def.17">PRALG_2:def 17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c16">gg</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:40_4_1"/><i><font color="Green" title="E45">A127</font></i>: 
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">gg</font>
 <b>and </b><br/><a NAME="E6:40_4_1"/><i><font color="Green" title="E46">A128</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">gg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span>
 <b>and </b><br/><a NAME="E7:40_4_1"/>
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span> holds <br/><font color="Maroon" title="c16">gg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="txt" href="#E2:40_4"><i><font color="Green" title="E37">A119</font></i></a>, <a class="txt" href="#E9:40_4"><i><font color="Green" title="E42">A124</font></i></a>, <a class="txt" href="#E2:40_4_1"><i><font color="Green" title="E43">A125</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">x1</font> = <font color="Maroon" title="c12">x1</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E5:40_4_1"><i><font color="Green" title="E45">A127</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c18">gg1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:40_4_1"/><i><font color="Green" title="E47">A129</font></i>: 
<font color="Maroon" title="c13">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">gg1</font>
 <b>and </b><br/><a NAME="E11:40_4_1"/><i><font color="Green" title="E48">A130</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">gg1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span>
 <b>and </b><br/><a NAME="E12:40_4_1"/>
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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span> holds <br/><font color="Maroon" title="c18">gg1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c15">s'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x'</font>
 <b>by </b><i><a class="txt" href="#E3:40_4"><i><font color="Green" title="E38">A120</font></i></a>, <a class="txt" href="#E9:40_4"><i><font color="Green" title="E42">A124</font></i></a>, <a class="txt" href="#E2:40_4_1"><i><font color="Green" title="E43">A125</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">x2</font> = <font color="Maroon" title="c13">x2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E10:40_4_1"><i><font color="Green" title="E47">A129</font></i></a></i>;<br/>
<a NAME="E14:40_4_1"/><i><font color="Green" title="E49">A131</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c17">x1</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="#E5:40_4_1"><i><font color="Green" title="E45">A127</font></i></a>, <a class="txt" href="#E6:40_4_1"><i><font color="Green" title="E46">A128</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E15:40_4_1"/><i><font color="Green" title="E50">A132</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c19">x2</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="#E10:40_4_1"><i><font color="Green" title="E47">A129</font></i></a>, <a class="txt" href="#E11:40_4_1"><i><font color="Green" title="E48">A130</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E16:40_4_1"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> )</span><br/>

<b>assume </b><a NAME="E1:40_4_1_1"/><i><font color="Green" title="E51">A133</font></i>: 
<font color="Maroon" title="c20">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font></span><br/>

<a NAME="E2:40_4_1_1"/><b>then </b><i><font color="Green" title="E52">A134</font></i>: 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c20">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <font color="Maroon" title="c4">EqR</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">f''</font> = <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">x2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="#E3:40_4"><i><font color="Green" title="E38">A120</font></i></a>, <a class="txt" href="#E9:40_4"><i><font color="Green" title="E42">A124</font></i></a>, <a class="txt" href="#E3:40_4_1"><i><font color="Green" title="E44">A126</font></i></a>, <a class="txt" href="pralg_3.html#E4"><i><font color="Green" title="E4">Lm1</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<font color="Maroon" title="c17">x1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c20">i</font></span>)</span> = 
<font color="Maroon" title="c21">f''</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:40_4"><i><font color="Green" title="E36">A118</font></i></a>, <a class="txt" href="#E2:40_4"><i><font color="Green" title="E37">A119</font></i></a>, <a class="txt" href="#E4:40_4"><i><font color="Green" title="E39">A121</font></i></a>, <a class="txt" href="#E7:40_4"><i><font color="Green" title="E40">A122</font></i></a>, <a class="txt" href="#E8:40_4"><i><font color="Green" title="E41">A123</font></i></a>, <a class="txt" href="#E9:40_4"><i><font color="Green" title="E42">A124</font></i></a>, <a class="txt" href="#E2:40_4_1_1"><i><font color="Green" title="E52">A134</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c19">x2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c20">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:40_4"><i><font color="Green" title="E38">A120</font></i></a>, <a class="txt" href="#E8:40_4"><i><font color="Green" title="E41">A123</font></i></a>, <a class="txt" href="#E9:40_4"><i><font color="Green" title="E42">A124</font></i></a>, <a class="txt" href="#E2:40_4_1_1"><i><font color="Green" title="E52">A134</font></i></a></i>
;<br/>
<b>then </b><font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">x2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <font color="Maroon" title="c4">EqR</font>,<font color="Maroon" title="c20">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="#E1:40_4_1_1"><i><font color="Green" title="E51">A133</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<font color="Maroon" title="c19">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
<b>by </b><i><a class="txt" href="#E1:40_4_1_1"><i><font color="Green" title="E51">A133</font></i></a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
;<br/>
<b>hence </b><a NAME="E6:40_4_1_1"/>
<font color="Maroon" title="c17">x1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">x2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:40_4_1"/>
<font color="Maroon" title="c12">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x2</font>
 <b>by </b><i><a class="txt" href="#E14:40_4_1"><i><font color="Green" title="E49">A131</font></i></a>, <a class="txt" href="#E15:40_4_1"><i><font color="Green" title="E50">A132</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E7:40"/>
<font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R4" title="MSUALG_3:pred.4">is_isomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_5"/>
<font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D12" title="MSUALG_3:def.12">MSUALG_3:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_5_1"/>
( <font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">F</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 <b>by </b><i><a class="txt" href="#E4:40"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="#E5:40"><i><font color="Green" title="E34">A71</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:40_5"/>
<font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R3" title="MSUALG_3:pred.3">is_monomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_5_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_5_2"/>
( <font color="Maroon" title="c9">F</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">F</font> is <a href="msualg_3.html#V1" title="MSUALG_3:attr.1">"1-1"</a> )
 <b>by </b><i><a class="txt" href="#E4:40"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="#E6:40"><i><font color="Green" title="E35">A117</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="msualg_3.html#D11" title="MSUALG_3:def.11">MSUALG_3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:40"/>
 <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">A</font>, <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <span class="p1">(<span class="default"><a href="pralg_3.html#K6" title="PRALG_3:func.6">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">A</font> <a href="pralg_3.html#K5" title="PRALG_3:func.5">/</a> <font color="Maroon" title="c4">EqR</font></span>)</span></span>)</span> <a href="msualg_3.html#R6" title="MSUALG_3:pred.6">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="msualg_3.html#D13" title="MSUALG_3:def.13">MSUALG_3:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
