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

<span class="kw">let </span><font color="Maroon" title="c1">DP</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">OAF</font> being    <a href="msalimit.html#M1" title="MSALIMIT:mode.1">OrderedAlgFam</a> of <font color="Maroon" title="c1">DP</font>,<font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">B</font> being   <a href="msalimit.html#V1" title="MSALIMIT:attr.1">normalized</a>   <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Olive" title="b2">OAF</font> holds   <a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Olive" title="b3">B</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="Olive" title="b2">OAF</font></span><br/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">OAF</font> being    <a href="msalimit.html#M1" title="MSALIMIT:mode.1">OrderedAlgFam</a> of <font color="Maroon" title="c1">DP</font>,<font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">B</font> being   <a href="msalimit.html#V1" title="MSALIMIT:attr.1">normalized</a>   <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Olive" title="b1">OAF</font> holds   <a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Olive" title="b2">B</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="Olive" title="b1">OAF</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">OAF</font> be    <a href="msalimit.html#M1" title="MSALIMIT:mode.1">OrderedAlgFam</a> of <font color="Maroon" title="c1">DP</font>,<font color="Maroon" title="c2">S</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being   <a href="msalimit.html#V1" title="MSALIMIT:attr.1">normalized</a>   <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Maroon" title="c3">OAF</font> holds   <a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Olive" title="b1">B</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="c3">OAF</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be   <a href="msalimit.html#V1" title="MSALIMIT:attr.1">normalized</a>   <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Maroon" title="c3">OAF</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</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="c3">OAF</font></span><br/>







<a NAME="E1:15"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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/> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</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 <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">a</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  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1"/>
<font color="Maroon" title="c5">a</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">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 <span class="p1">(<span class="default"><a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">s</font> = <font color="Maroon" title="c5">a</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">thus </span><a NAME="E3:15_1"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
  <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">e</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="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c7">e</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<a NAME="E1:15_1_1"/>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</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">OAF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#D9" onmouseover="rs('msualg_2/D9')" onmouseout="rh()">MSUALG_2:def 9</a></span>;<br/>
<a NAME="E2:15_1_1"/><span class="kw">then </span>
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</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">OAF</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" onmouseover="rs('pboole/D18')" onmouseout="rh()">PBOOLE:def 18</a></span>;<br/>
<a NAME="E3:15_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D2" onmouseover="rs('pboole/D2')" onmouseout="rh()">PBOOLE:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:15_1_1"/>
<font color="Maroon" title="c7">e</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="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">e</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">hence </span><a NAME="E5:15_1_1"/>
<font color="Maroon" title="c7">e</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:15_1_1"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">e</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> or <font color="Maroon" title="c7">e</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="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:15_1"/>
<font color="Maroon" title="c7">e</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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">e</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="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="pralg_2.html#K10" title="PRALG_2:func.10">SORTS</a> <font color="Maroon" title="c3">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#T12" onmouseover="rs('pralg_2/T12')" onmouseout="rh()">PRALG_2:12</a></span>;<br/>
<a NAME="E6:15_1"/>
 for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">DP</font>  st <font color="Olive" title="b1">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="15_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">i</font>, <font color="Maroon" title="c10">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">DP</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">j</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:15_1_2"/>
<font color="Maroon" title="c9">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c10">j</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font></span><br/>

<a NAME="E2:15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#T1" onmouseover="rs('orders_3/T1')" onmouseout="rh()">ORDERS_3:1</a></span>;<br/>
<a NAME="E3:15_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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">OAF</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 
;<br/>
<a NAME="E4:15_1_2"/><span class="kw">then </span>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">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">OAF</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <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="c1">DP</font> &amp; <font color="Maroon" title="c8">f</font> <a href="tarski.html#R2" title="TARSKI: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">OAF</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a>, <a class="ref" href="pralg_2.html#D10" onmouseover="rs('pralg_2/D10')" onmouseout="rh()">PRALG_2:def 10</a></span>;<br/>
<a NAME="E5:15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="tarski.html#R2" title="TARSKI: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">OAF</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T9" onmouseover="rs('card_3/T9')" onmouseout="rh()">CARD_3:9</a></span>;<br/>
 <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c9">i</font>) = 
<font color="Maroon" title="c4">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c9">i</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="msalimit.html#D3" target="_self" onmouseover="rs('msalimit/D3')" onmouseout="rh()">Def3</a>, <a class="ref" href="orders_2.html#T1" onmouseover="rs('orders_2/T1')" onmouseout="rh()">ORDERS_2:1</a></span>
<br/>.= 
 <a href="msualg_3.html#K2" title="MSUALG_3:func.2">id</a>  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c9">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msalimit.html#D4" target="_self" onmouseover="rs('msalimit/D4')" onmouseout="rh()">Def4</a></span>
;<br/>
<a NAME="E7:15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<span class="p1">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c9">i</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#D1" onmouseover="rs('msualg_3/D1')" onmouseout="rh()">MSUALG_3:def 1</a></span>;<br/>
<a NAME="E8:15_1_2"/>
 ex <font color="Olive" title="b1">U0</font> being    <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> over <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">U0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">OAF</font> <a href="pralg_2.html#K5" title="PRALG_2:func.5">.</a> <font color="Maroon" title="c9">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">OAF</font>,<font color="Maroon" title="c6">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="Olive" title="b1">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_2.html#D9" onmouseover="rs('pralg_2/D9')" onmouseout="rh()">PRALG_2:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:15_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c10">j</font>)</span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E7:15_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:15_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:15_1"/>
<font color="Maroon" title="c7">e</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="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msalimit.html#D6" target="_self" onmouseover="rs('msalimit/D6')" onmouseout="rh()">Def6</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:15"/>
 <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">OAF</font> is    <a href="msualg_2.html#M1" title="MSUALG_2:mode.1">MSSubAlgebra</a> of  <a href="pralg_2.html#K14" title="PRALG_2:func.14">product</a> <font color="Maroon" title="c3">OAF</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T5" onmouseover="rs('msualg_2/T5')" onmouseout="rh()">MSUALG_2:5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:15"/>
 <a href="msalimit.html#K3" title="MSALIMIT:func.3">InvLim</a> <font color="Maroon" title="c4">B</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="c3">OAF</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="msualg_2.html#T9" onmouseover="rs('msualg_2/T9')" onmouseout="rh()">MSUALG_2:9</a>, <a class="ref" href="pboole.html#T3" onmouseover="rs('pboole/T3')" onmouseout="rh()">PBOOLE:3</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
