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<span class="kw">let </span><font color="Maroon" title="c1">R</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" 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#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">F</font> is  <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a>  holds <br/><font color="Olive" title="b2">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Olive" title="b1">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</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">F</font> being  <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b1">F</font> is  <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a>  holds <br/><font color="Olive" title="b1">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</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="c3">F</font> is  <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a>  implies <font color="Maroon" title="c3">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> 
 ; <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="c3">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">C</font> = <font color="Maroon" title="c3">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">C1</font> = <font color="Maroon" title="c3">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E3:10"/>
<font color="Maroon" title="c5">C1</font> is  <a href="osalg_1.html#V11" title="OSALG_1:attr.11">order-sorted</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</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="c6">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s2</font> or <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_1"/>
<font color="Maroon" title="c6">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">s2</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="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font></span><br/>

<a NAME="E2:10_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="ref" href="osalg_3.html#T2" target="_self" title="OSALG_3:th.2">Th2</a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></span>;<br/>
<a NAME="E3:10_1"/>
( <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font></span>)</span> &amp; <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D25" title="PBOOLE:def.25">PBOOLE:def 25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:10_1"/>
<font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:10_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="relat_1.html#T158" title="RELAT_1:th.158">RELAT_1:158</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>
<span class="kw">hence </span><a NAME="E4:10"/>
<font color="Maroon" title="c3">F</font> <a href="pboole.html#K11" title="PBOOLE:func.11">.:.:</a> <font color="Maroon" title="c2">A</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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