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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being   <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s</font> holds <br/><font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Olive" title="b2">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">A</font> be    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Olive" title="b1">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <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>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font></span>)</span></span><br/>

<a NAME="E2:22"/>
<font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font>
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<a NAME="E3:22"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> where <font color="Olive" title="b1">s1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E4:22"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="osalg_4.html#K4" title="OSALG_4:func.4">-carrier_of</a> <span class="p1">(<span class="default"><a href="osalg_4.html#K3" title="OSALG_4:func.3">CComp</a> <font color="Maroon" title="c3">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</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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