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<b>let </b><font color="Maroon" title="c1">I</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">J</font> be  <a href="waybel_3.html#V4" title="WAYBEL_3:attr.4">non-Empty</a> <a href="waybel18.html#V1" title="WAYBEL18:attr.1">TopSpace-yielding</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:18"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel18.html#K2" title="WAYBEL18:func.2">product_prebasis</a> <font color="Maroon" title="c2">J</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c5">T</font> being    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> , <font color="Maroon" title="c6">Ai</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T</font><b> such that </b><br/><a NAME="E3:18"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon" title="c6">Ai</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c5">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">J</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <font color="Maroon" title="c2">J</font></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">Ai</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#D2" title="WAYBEL18:def.2">WAYBEL18:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">i</font> = <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="yellow17.html#E3:18"><i><font color="Green" title="E18">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">i</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c8">Ai</font> = <font color="Maroon" title="c6">Ai</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">J</font> <a href="waybel18.html#K4" title="WAYBEL18:func.4">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <b>by </b><i><a class="txt" href="yellow17.html#E5:18"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">Ai</font>
;<br/>

<b>thus </b><a NAME="E9:18"/>
<font color="Maroon" title="c8">Ai</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="yellow17.html#E4:18"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="yellow17.html#E5:18"><i><font color="Green" title="E20">A3</font></i></a></i>;<br/>

<b>thus </b><a NAME="E10:18"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K6" title="WAYBEL18:func.6">proj</a> <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c7">i</font></span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c8">Ai</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="yellow17.html#E6:18"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="waybel18.html#T5" title="WAYBEL18:th.5">WAYBEL18:5</a></i>;<br/>


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