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<span class="kw">let </span><font color="Maroon" title="c1">S0</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">s</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">ol</font> =  the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span>;<br/>
<a NAME="E1:19"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S0</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 <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span> &amp;  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D5" target="_self" title="OSALG_1:def.5">Def6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:19"/>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font> is  <a href="orders_3.html#V1" title="ORDERS_3:attr.1">discrete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_3.html#D1" title="ORDERS_3:def.1">ORDERS_3:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a> </span><br/>

<a NAME="E3:19"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D5" target="_self" title="OSALG_1:def.5">Def6</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1"/>
<font color="Maroon" title="c4">x</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font></span><br/><a NAME="E2:19_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#D2" target="_self" title="OSALG_1:def.2">Def3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:19_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="relat_1.html#D10" title="RELAT_1:def.10">RELAT_1:def 10</a></span>; <a class="txt" 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="E5:19"/>
 <a href="osalg_1.html#K1" title="OSALG_1:func.1">OSSign</a> <font color="Maroon" title="c1">S0</font> is  <a href="osalg_1.html#V6" title="OSALG_1:attr.6">op-discrete</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T3" target="_self" title="OSALG_1:th.3">Th3</a></span>; <a class="txt" 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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