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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">S</font>, <font color="Olive" title="b2">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">L</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">T</font> holds <br/>for <font color="Olive" title="b3">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">S</font> holds <br/> ( <font color="Olive" title="b3">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">T</font> &amp; ( for <font color="Olive" title="b4">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">T</font>  st <font color="Olive" title="b3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">Y</font> holds <br/>( ( <font color="Olive" title="b3">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Olive" title="b4">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Olive" title="b3">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Olive" title="b4">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">S1</font>, <font color="Maroon" title="c3">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</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"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> implies for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font> holds <br/> ( <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> &amp; ( for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/>( ( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Olive" title="b2">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Olive" title="b2">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) ) ) )</span><br/>



<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E3">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</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="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</font> holds <br/> ( <font color="Olive" title="b1">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> &amp; ( for <font color="Olive" title="b2">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font> holds <br/>( ( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Olive" title="b2">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Olive" title="b2">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) ) )</span><br/>

<b>let </b><font color="Maroon" title="c4">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S1</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> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font> &amp; ( for <font color="Olive" title="b1">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>  st <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> holds <br/>( ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Olive" title="b1">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Olive" title="b1">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) ) )</span><br/>

<b>thus </b><a NAME="E2:8"/>
<font color="Maroon" title="c4">X</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>
 <b>by </b><i><a class="txt" href="yellow_2.html#E1:8"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <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">Y</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</font>  st <font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">Y</font> holds <br/>( ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Olive" title="b1">Y</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Olive" title="b1">Y</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) )</span><br/>

<b>let </b><font color="Maroon" title="c5">X2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">S2</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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X2</font> implies ( ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) ) )</span><br/>

<b>assume </b><a NAME="E3:8"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X2</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> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> implies <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> ) &amp; ( <font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> ) )</span><br/>

<div><b>hereby </b> <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> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> implies <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green" title="E5">A3</font></i>: 
<font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a></span><br/><b>thus </b><a NAME="E2:8_1"/>
<font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> ) )</span><br/>

<b>assume </b><a NAME="E1:8_1_1"/><i><font color="Green" title="E6">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font><b> such that </b><br/><a NAME="E4:8_1_1"/><i><font color="Green" title="E7">A5</font></i>: 
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Maroon" title="c10">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">y'</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E1:8_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="txt" href="yellow_2.html#E1:8_1_1"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="waybel_0.html#D2" title="WAYBEL_0:def.2">WAYBEL_0:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c13">z1</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E4:8_1_1"><i><font color="Green" title="E7">A5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )</span><br/>

<a NAME="E7:8_1_1"/>
( <font color="Maroon" title="c13">z1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">x1</font> &amp; <font color="Maroon" title="c13">z1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">y1</font> )
 
<b>by </b><i><a class="txt" href="yellow_2.html#E4:8_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<b>hence </b><a NAME="E8:8_1_1"/>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E4:8_1_1"><i><font color="Green" title="E7">A5</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E5:8"/><i><font color="Green" title="E5">A6</font></i>: 
<font color="Maroon" title="c4">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">X2</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a></span><br/>

<b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  not <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) )</span><br/>

<b>assume </b><a NAME="E6:8"/><i><font color="Green" title="E6">A7</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">S2</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y'</font> = <font color="Maroon" title="c7">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font> <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S1</font><b> such that </b><br/><a NAME="E9:8"/><i><font color="Green" title="E7">A8</font></i>: 
( <font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">x'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z</font> &amp; <font color="Maroon" title="c9">y'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E5:8"><i><font color="Green" title="E5">A6</font></i></a>, <a class="txt" href="yellow_2.html#E6:8"><i><font color="Green" title="E6">A7</font></i></a>, <a class="ref" href="waybel_0.html#D1" title="WAYBEL_0:def.1">WAYBEL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x1</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c12">y1</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c13">z1</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c10">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">S2</font> <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E9:8"><i><font color="Green" title="E7">A8</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">z</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> )</span><br/>

<a NAME="E12:8"/>
( <font color="Maroon" title="c11">x1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">z1</font> &amp; <font color="Maroon" title="c12">y1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c13">z1</font> )
 
<b>by </b><i><a class="txt" href="yellow_2.html#E9:8"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<b>hence </b><a NAME="E13:8"/>
( <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> )
 <b>by </b><i><a class="txt" href="yellow_2.html#E3:8"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="yellow_2.html#E9:8"><i><font color="Green" title="E7">A8</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</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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