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<b>let </b><font color="Maroon" title="c1">S</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</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">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</a> of <font color="Maroon" title="c1">S</font> iff  ex <font color="Olive" title="b2">op1</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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Olive" title="b2">op1</font> )</span><br/><b>let </b><font color="Maroon" title="c2">X</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="c2">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</a> of <font color="Maroon" title="c1">S</font> iff  ex <font color="Olive" title="b1">op1</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="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Olive" title="b1">op1</font> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">op1</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="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Olive" title="b1">op1</font> implies <font color="Maroon" title="c2">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</a> of <font color="Maroon" title="c1">S</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:88_1"/>
<font color="Maroon" title="c2">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</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">  ex <font color="Olive" title="b1">x1</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="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Olive" title="b1">x1</font></span><br/><b>then consider </b><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> the <a href="osalg_1.html#U1" title="OSALG_1:sel.1">Overloading</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c3">x</font> )
 <b>by </b><i><a class="ref" href="eqrel_1.html#D5" title="EQREL_1:def.5">EQREL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="#E3:88_1"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c4">x1</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">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Maroon" title="c5">x1</font></span><br/><b>thus </b><a NAME="E5:88_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Maroon" title="c5">x1</font>
 <b>by </b><i><a class="txt" href="#E3:88_1"><i><font color="Green" title="E48">A1</font></i></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>given </b><font color="Maroon" title="c3">op1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E3:88"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K10" title="OSALG_1:func.10">Name</a> <font color="Maroon" title="c3">op1</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">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</a> of <font color="Maroon" title="c1">S</font></span><br/>

<b>thus </b><a NAME="E4:88"/>
<font color="Maroon" title="c2">X</font> is   <a href="osalg_1.html#NM6" title="OSALG_1:NM.6">OperName</a> of <font color="Maroon" title="c1">S</font>
 <b>by </b><i><a class="txt" href="#E3:88"><i><font color="Green" title="E48">A2</font></i></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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