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<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
 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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">thus </span><a NAME="E2:6"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">let </span><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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:6"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">assume </span><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E5">A3</font></span>: 
<font color="Maroon" title="c4">X</font> is  <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">thus </span><a NAME="E2:6_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D2" onmouseover="rs('waybel_0/D2')" onmouseout="rh()">WAYBEL_0:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  not <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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/>

<span class="kw">assume </span><a NAME="E1:6_1_1"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tarski.html#R2" title="TARSKI: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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:6_1_1"/><span class="lab"><font color="Green" title="E7">A5</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E5:6_1_1"/><span class="lab"><font color="Green" title="E8">A6</font></span>: 
( <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">x9</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">y9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E5">A3</font></span></a>, <a class="txt" href="#E1:6_1_1"><span class="lab"><font color="Green" title="E6">A4</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T58" onmouseover="rs('yellow_0/T58')" onmouseout="rh()">YELLOW_0:58</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <a href="tarski.html#R2" title="TARSKI: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="E8:6_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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:6_1_1"><span class="lab"><font color="Green" title="E8">A6</font></span></a>, <a class="ref" href="yellow_0.html#T59" onmouseover="rs('yellow_0/T59')" onmouseout="rh()">YELLOW_0:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:6_1_1"/>
( <font color="Maroon" title="c14">z</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E4:6_1_1"><span class="lab"><font color="Green" title="E7">A5</font></span></a>, <a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:6"/><span class="lab"><font color="Green" title="E5">A7</font></span>: 
<font color="Maroon" title="c4">X</font> is  <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="waybel_0.html#D1" onmouseover="rs('waybel_0/D1')" onmouseout="rh()">WAYBEL_0:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> or  not <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI: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="tarski.html#R2" title="TARSKI: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/>

<span class="kw">assume </span><a NAME="E6:6"/><span class="lab"><font color="Green" title="E6">A8</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> &amp; <font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X2</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="tarski.html#R2" title="TARSKI: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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">y9</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>;<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:6"/><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<font color="Maroon" title="c10">z</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <span class="kw">and </span><br/><a NAME="E10:6"/><span class="lab"><font color="Green" title="E8">A10</font></span>: 
( <font color="Maroon" title="c8">x9</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">y9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E5:6"><span class="lab"><font color="Green" title="E5">A7</font></span></a>, <a class="txt" href="#E6:6"><span class="lab"><font color="Green" title="E6">A8</font></span></a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T58" onmouseover="rs('yellow_0/T58')" onmouseout="rh()">YELLOW_0:58</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E7">A9</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c14">z</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c14">z</font> <a href="tarski.html#R2" title="TARSKI: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="E13:6"/>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:6"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="ref" href="yellow_0.html#T59" onmouseover="rs('yellow_0/T59')" onmouseout="rh()">YELLOW_0:59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:6"/>
( <font color="Maroon" title="c14">z</font> <a href="tarski.html#R2" title="TARSKI: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 class="kw">by</span> <span class="lab"><a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E4">A2</font></span></a>, <a class="txt" href="#E9:6"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="ref" href="yellow_0.html#T60" onmouseover="rs('yellow_0/T60')" onmouseout="rh()">YELLOW_0:60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
