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<b>let </b><font color="Maroon">L1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="waybel_0.html#V24">up-complete</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">L2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>







<b>assume </b><a NAME="E1:16"/><i><font color="Green">E34</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L1</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L2</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L2</font> #) &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E3:16_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> )
 ;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/><a NAME="E5:16_1_1"/>
<font color="Maroon">L2</font> is <a href="orders_2.html#V4">antisymmetric</a>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T14">WAYBEL_8:14</a></i>;<br/><a NAME="E6:16_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E2:16_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">y</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E4:16_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font> )
 ;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/><a NAME="E6:16_1"/>
( <font color="Maroon">L2</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L2</font> is <a href="orders_2.html#V4">antisymmetric</a> )
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T14">WAYBEL_8:14</a>, <a class="ref" href="waybel_8.html#T15">WAYBEL_8:15</a></i>;<br/><a NAME="E7:16_1"/><b>then </b>
<font color="Maroon">w</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E8:16_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:16_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E3:16_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> <b>by </b><i/>;<br/><a NAME="E5:16_2"/>
<font color="Maroon">L2</font> is <a href="orders_2.html#V4">antisymmetric</a>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#T14">WAYBEL_8:14</a></i>;<br/><a NAME="E6:16_2"/><b>then </b>
<font color="Maroon">w</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/><b>hence </b><a NAME="E7:16_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">a</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="E4:16"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E6:16"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">y</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> <b>by </b><i/>;<br/>
<a NAME="E8:16"/>
( <font color="Maroon">L2</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L2</font> is <a href="orders_2.html#V4">antisymmetric</a> )
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#T14">WAYBEL_8:14</a>, <a class="ref" href="waybel_8.html#T15">WAYBEL_8:15</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<font color="Maroon">w</font> <a href="waybel_3.html#NR3" target="_self">&gt;&gt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_8.html#T8">WAYBEL_8:8</a></i>;<br/>
<b>hence </b><a NAME="E10:16"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


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