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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="struct_0.html#L1">1-sorted</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">W</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K3">BoolePoset</a> 1;<br/>





<a NAME="E1:70"/><i><font color="Green">E29</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K3">BoolePoset</a> 1
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">c1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>, <font color="Maroon">c2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E2:70"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c1</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">c2</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Maroon">W</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:70_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">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">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:70_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <b>by </b><i/>;<br/><a NAME="E4:70_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) or  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/><a NAME="E5:70_1_1"/><b>then </b>
( ( <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 ) or <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><a NAME="E6:70_1_1"/><b>then </b>
<font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E7:70_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/><a NAME="E8:70_1_1"/><b>then </b>
<font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/><b>hence </b><a NAME="E9:70_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:70_1"/>
<font color="Maroon">c1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c2</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:70"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c1</font> <a href="yellow_2.html#R1">&lt;=</a> <font color="Maroon">c2</font>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E5:70"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>, <font color="Maroon">b</font> = <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E8:70"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Olive">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_2.html#D1">YELLOW_2:def 1</a></i>;<br/>
<a NAME="E9:70"/><b>then </b>
( <font color="Maroon">c1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c2</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/>
<a NAME="E10:70"/><b>then </b>
( 1 <a href="tarski.html#R1">c=</a> 0 &amp; 0 <a href="tarski.html#R1">c=</a> 1 &amp; 0 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a>, <a class="ref" href="yellow_1.html#T2">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">XBOOLE_0:def 10</a></i>;<br/>


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