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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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">P</font>,<font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">B</font> being    <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Olive" title="b2">OAF</font><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b4">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b4">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">j</font> holds <br/><font color="Olive" title="b3">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Olive" title="b3">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">i</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">P</font>,<font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">B</font> being    <a href="msalimit.html#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Olive" title="b1">OAF</font><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b3">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">j</font> holds <br/><font color="Olive" title="b2">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">j</font>,<font color="Olive" title="b3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Olive" title="b2">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b4">j</font>,<font color="Olive" title="b3">i</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">P</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#M2" title="MSALIMIT:mode.2">Binding</a> of <font color="Maroon" title="c3">OAF</font><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</font>  st <font color="Olive" title="b2">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Olive" title="b1">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b3">j</font>,<font color="Olive" title="b2">i</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">B</font> be    <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">  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">P</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> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c4">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">j</font>,<font color="Olive" title="b1">i</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">P</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="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> implies <font color="Maroon" title="c4">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:10"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">and </span><br/><a NAME="E2:10"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">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"> <font color="Maroon" title="c4">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>)</span><br/>

<a NAME="E3:10"/>
( <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><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="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="c5">i</font></span>)</span></span>)</span>,<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="c5">i</font>,<font color="Maroon" title="c6">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><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="p3">(<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="c5">i</font></span>)</span></span>)</span></span>)</span>) &amp;  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>,<span class="p1">(<span class="default"><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="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="c5">i</font></span>)</span></span>)</span>,<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="c5">i</font>,<font color="Maroon" title="c6">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p2">(<span class="default"><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="p3">(<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="c5">i</font></span>)</span></span>)</span></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c5">i</font>,<font color="Maroon" title="c6">j</font>)</span>)</span> <a href="msualg_3.html#K3" title="MSUALG_3:func.3">**</a> <span class="p1">(<span class="default"><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="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="c5">i</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:10"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="msalimit.html#D5" target="_self" onmouseover="rs('msalimit/D5')" onmouseout="rh()">Def5</a>, <a class="ref" href="funcop_1.html#D8" onmouseover="rs('funcop_1/D8')" onmouseout="rh()">FUNCOP_1:def 8</a></span>;<br/>
<a NAME="E4:10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msalimit.html#K1" title="MSALIMIT:func.1">bind</a> (<font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T3" onmouseover="rs('msualg_3/T3')" onmouseout="rh()">MSUALG_3:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:10"/>
<font color="Maroon" title="c4">B</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msalimit.html#K2" title="MSALIMIT:func.2">Normalized</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="msalimit.html#D3" target="_self" onmouseover="rs('msalimit/D3')" onmouseout="rh()">Def3</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/>


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