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<b>let </b><font color="Maroon" title="c1">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="osalg_1.html#L1" title="OSALG_1:struct.1">OverloadedMSSign</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">o</font>, <font color="Olive" title="b2">o1</font>, <font color="Olive" title="b3">o2</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b2">o1</font> &amp; <font color="Olive" title="b2">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b3">o2</font> holds <br/><font color="Olive" title="b1">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Olive" title="b3">o2</font></span><br/>

<a NAME="E1:15"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="orders_1.html#T97" title="ORDERS_1:th.97">ORDERS_1:97</a></i>;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E6">A1</font></i>: 
the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <font color="Maroon" title="c1">S</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<b>let </b><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">o1</font>, <font color="Maroon" title="c4">o2</font> be   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">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="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font> implies <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font> )</span><br/>

<b>assume </b><a NAME="E3:15"/>
( <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c3">o1</font> &amp; <font color="Maroon" title="c3">o1</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font></span><br/>

<a NAME="E4:15"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">o1</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 <font color="Maroon" title="c1">S</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">o1</font>,<font color="Maroon" title="c4">o2</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 <font color="Maroon" title="c1">S</font> )
 
<b>by </b><i><a class="ref" href="osalg_1.html#D3" target="_self" title="OSALG_1:def.3">Def3</a></i>;<br/>
<a NAME="E5:15"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">o2</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 <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="#E2:15"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="relat_2.html#D8" title="RELAT_2:def.8">RELAT_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:15"/>
<font color="Maroon" title="c2">o</font> <a href="osalg_1.html#R1" title="OSALG_1:pred.1">~=</a> <font color="Maroon" title="c4">o2</font>
 <b>by </b><i><a class="ref" href="osalg_1.html#D3" target="_self" title="OSALG_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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