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<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">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Olive" title="b1">S</font><br/> for <font color="Olive" title="b4">a</font> being 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><br/> for <font color="Olive" title="b5">Aa</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b4">a</font>,<font color="Olive" title="b1">S</font>  st <font color="Olive" title="b2">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">Aa</font> holds <br/> <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Olive" title="b5">Aa</font>,<font color="Olive" title="b3">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="Olive" title="b2">A</font>,<font color="Olive" title="b3">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b4">a</font></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">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b3">a</font> being 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><br/> for <font color="Olive" title="b4">Aa</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b3">a</font>,<font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Aa</font> holds <br/> <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Olive" title="b4">Aa</font>,<font color="Olive" title="b2">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="Olive" title="b1">A</font>,<font color="Olive" title="b2">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b3">a</font></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">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font><br/> for <font color="Olive" title="b2">a</font> being 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><br/> for <font color="Olive" title="b3">Aa</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b2">a</font>,<font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">Aa</font> holds <br/> <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Olive" title="b3">Aa</font>,<font color="Olive" title="b1">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="Olive" title="b1">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b2">a</font></span><br/><b>let </b><font color="Maroon" title="c4">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</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">a</font> being 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><br/> for <font color="Olive" title="b2">Aa</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Olive" title="b1">a</font>,<font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Aa</font> holds <br/> <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Olive" title="b2">Aa</font>,<font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Olive" title="b1">a</font></span><br/><b>let </b><font color="Maroon" title="c5">a</font> be  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>; <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">Aa</font> being   <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Aa</font> holds <br/> <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Olive" title="b1">Aa</font>,<font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font></span><br/><b>let </b><font color="Maroon" title="c6">Aa</font> be    <a href="pralg_2.html#M2" title="PRALG_2:mode.2">MSAlgebra-Family</a> of <font color="Maroon" title="c5">a</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"> ( <font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Aa</font> implies  <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font> )</span><br/>











<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Aa</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#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font></span><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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</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="c4">s</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c5">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="c5">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="c5">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="c7">Cas</font> = <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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c5">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="E4:5"/>
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="c5">a</font> holds <br/><span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">s</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="c7">Cas</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="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> implies <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Cas</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> )</span><br/>

<b>assume </b><a NAME="E1:5_2"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</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_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Cas</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c9">i'</font> = <font color="Maroon" title="c8">i</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">a</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i''</font> = <font color="Maroon" title="c9">i'</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ;<br/>
<a NAME="E4:5_2"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c6">Aa</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c9">i'</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="c9">i'</font>
 
<b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">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="E6:5_2"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c11">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Aa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> &amp; <span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c11">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s</font> )
 <b>by </b><i><a class="txt" href="#E1:5_2"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="pralg_2.html#D16" title="PRALG_2:def.16">PRALG_2:def 16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">U1</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="E8:5_2"/><i><font color="Green" title="E7">A5</font></i>: 
( <font color="Maroon" title="c12">U1</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="c10">i''</font> &amp; <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="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c12">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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/>
<b>thus </b><a NAME="E9:5_2"/>
<span class="p1">(<span class="default"><a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Cas</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font>
 <b>by </b><i><a class="txt" href="#E4:5_2"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="#E6:5_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="#E8:5_2"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="E5:5"/>
 <a href="pralg_2.html#K9" title="PRALG_2:func.9">Carrier</a> <font color="Maroon" title="c6">Aa</font>,<font color="Maroon" title="c4">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="c4">s</font></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
