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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ;<br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">W</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c4">S</font> be    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1;<br/>





<a NAME="E1:70"/><i><font color="Green" title="E42">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c5">c1</font>, <font color="Maroon" title="c6">c2</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c4">S</font>;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c5">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c2">V</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c6">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <font color="Maroon" title="c3">W</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">W</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon" title="c7">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">z</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> <b>by </b><i><a class="txt" href="yellow16.html#E1:70"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/><a NAME="E4:70_1_1"/>
( ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> ) or  not <font color="Maroon" title="c8">x</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="yellow16.html#E1:70_1"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/><a NAME="E5:70_1_1"/><b>then </b>
( ( <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) or <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="txt" href="yellow16.html#E2:70"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/><a NAME="E6:70_1_1"/><b>then </b>
<font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><a NAME="E7:70_1_1"/><b>then </b>
<font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/><a NAME="E8:70_1_1"/><b>then </b>
<font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="yellow16.html#E1:70"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/><b>hence </b><a NAME="E9:70_1_1"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:70_1"/>
<font color="Maroon" title="c5">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c2</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:70"/><i><font color="Green" title="E44">A4</font></i>: 
<font color="Maroon" title="c5">c1</font> <a href="yellow_2.html#R1" title="YELLOW_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c2</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">x</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><br/>

<b>assume </b><a NAME="E5:70"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">W</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">a</font> = <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> 1</span>)</span> <b>by </b><i><a class="txt" href="yellow16.html#E1:70"><i><font color="Green" title="E42">A1</font></i></a></i>;<br/>
<a NAME="E8:70"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> &amp; <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">b</font> )
 
<b>by </b><i><a class="txt" href="yellow16.html#E4:70"><i><font color="Green" title="E44">A4</font></i></a>, <a class="ref" href="yellow_2.html#D1" title="YELLOW_2:def.1">YELLOW_2:def 1</a></i>;<br/>
<a NAME="E9:70"/><b>then </b>
( <font color="Maroon" title="c5">c1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">c2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c9">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c10">b</font> )
 
<b>by </b><i><a class="txt" href="yellow16.html#E1:70"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="yellow16.html#E2:70"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="yellow16.html#E5:70"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
( 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 0 &amp; 0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 &amp; 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a>, <a class="ref" href="yellow_1.html#T2" title="YELLOW_1:th.2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E11:70"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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