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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">V</font>, <font color="Olive" title="b2">S</font>, <font color="Olive" title="b3">T</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">S</font> <a href="yellow_4.html#R1" title="YELLOW_4:pred.1">is_finer_than</a> <font color="Olive" title="b3">T</font> &amp; <font color="Olive" title="b1">V</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> &amp; <font color="Olive" title="b3">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Olive" title="b2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">S</font>, <font color="Maroon" title="c4">T</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">S</font> <a href="yellow_4.html#R1" title="YELLOW_4:pred.1">is_finer_than</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c2">V</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> &amp; <font color="Maroon" title="c4">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> implies <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="yellow_4.html#R1" title="YELLOW_4:pred.1">is_finer_than</a> <font color="Maroon" title="c4">T</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c4">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">V</font></span><br/>

<b>let </b><font color="Maroon" title="c5">q</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> )</span><br/>

<b>assume </b><a NAME="E4:23"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">S1</font> = <font color="Maroon" title="c3">S</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c5">q</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c6">S1</font> <b>by </b><i><a class="txt" href="#E4:23"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E8:23"/><i><font color="Green" title="E17">A5</font></i>: 
( <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c7">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">b</font> )
 <b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="yellow_4.html#D1" title="YELLOW_4:def.1">YELLOW_4:def 1</a></i>;<br/>
<b>thus </b><a NAME="E9:23"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <b>by </b><i><a class="txt" href="#E2:23"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="#E3:23"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="#E8:23"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="waybel_0.html#D19" title="WAYBEL_0:def.19">WAYBEL_0:def 19</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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