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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a>  <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">E</font> being  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of <font color="Olive" title="b1">A</font><br/> for <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b3">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b4">s2</font> holds <br/> <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Olive" title="b2">E</font>,<font color="Olive" title="b3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Olive" title="b2">E</font>,<font color="Olive" title="b4">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">E</font> being  <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of <font color="Maroon" title="c2">A</font><br/> for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s2</font> holds <br/> <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Olive" title="b1">E</font>,<font color="Olive" title="b2">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Olive" title="b1">E</font>,<font color="Olive" title="b3">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">E</font> be   <a href="msualg_4.html#V3" title="MSUALG_4:attr.3">MSEquivalence-like</a>   <a href="osalg_4.html#M1" title="OSALG_4:mode.1">OrderSortedRelation</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> holds <br/> <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Olive" title="b1">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Olive" title="b2">s2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">s2</font> implies  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font> )</span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c6">s3</font> = <font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c7">s4</font> = <font color="Maroon" title="c5">s2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<span class="kw">assume </span><a NAME="E2:35"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c4">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">s2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font></span><br/>

<a NAME="E3:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c5">s2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_4.html#T5" target="_self" title="OSALG_4:th.5">Th5</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:35"/>
 <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">SO</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c2">A</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">s1</font> or <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:35_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c4">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font></span><br/>

<a NAME="E3:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</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 <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> &amp; <font color="Maroon" title="c9">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K5" title="OSALG_4:func.5">CompClass</a> <font color="Maroon" title="c3">E</font>,<span class="p2">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c4">s1</font></span>)</span></span>)</span>,<font color="Olive" title="b1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_4.html#D12" target="_self" title="OSALG_4:def.12">Def12</a></span>;<br/>
<a NAME="E4:35_1"/>
<font color="Maroon" title="c8">SO</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s3</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">SO</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:35_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K6" title="OSALG_4:func.6">OSClass</a> <font color="Maroon" title="c3">E</font>,<font color="Maroon" title="c5">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:35_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="osalg_4.html#D12" target="_self" title="OSALG_4:def.12">Def12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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


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