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<b>let </b><font color="Maroon" title="c1">S</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> ;<br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>;<br/>



<b>set </b><font color="Maroon" title="c4">f</font> =  <a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font>;<br/>
<b>thus </b><a NAME="E1:47"/>
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c3">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K3" title="YELLOW_4:func.3">"/\"</a> <font color="Maroon" title="c3">D2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</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="E1:47_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c3">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">q1</font> = <font color="Maroon" title="c5">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c1">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span><b> such that </b><br/><a NAME="E4:47_1"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c7">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c3">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E1:47_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:47_1"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="waybel_2.html#E4:47_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c11">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="waybel_2.html#E6:47_1"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<font color="Maroon" title="c5">q</font> = 
<span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="txt" href="waybel_2.html#E4:47_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="waybel_2.html#E6:47_1"><i><font color="Green" title="E32">A3</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c10">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c11">y</font>
<b>by </b><i><a class="ref" href="waybel_2.html#D4" target="_self" title="WAYBEL_2:def.4">Def4</a></i>
;<br/>
<a NAME="E9:47_1"/><b>then </b>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">b</font></span>)</span> where <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel_2.html#E6:47_1"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:47_1"/>
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K3" title="YELLOW_4:func.3">"/\"</a> <font color="Maroon" title="c3">D2</font>
 <b>by </b><i><a class="ref" href="yellow_4.html#D4" title="YELLOW_4:def.4">YELLOW_4:def 4</a></i>;<br/>


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

<b>reconsider </b><font color="Maroon" title="c5">X</font> = <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c3">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<b>let </b><font color="Maroon" title="c6">q</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="E3:47"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> <a href="yellow_4.html#K3" title="YELLOW_4:func.3">"/\"</a> <font color="Maroon" title="c3">D2</font>
 ;<br/>

<a NAME="E4:47"/><b>then </b>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Olive" title="b2">y</font></span>)</span> where <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> : ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow_4.html#D4" title="YELLOW_4:def.4">YELLOW_4:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E6:47"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c8">y</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D2</font> )
 ;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E6:47"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="waybel_2.html#D4" target="_self" title="WAYBEL_2:def.4">Def4</a></i>;<br/>
<a NAME="E8:47"/>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="waybel_2.html#E6:47"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E9:47"/>
<font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K4" title="WAYBEL_2:func.4">inf_op</a> <font color="Maroon" title="c1">S</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c2">D1</font>,<font color="Maroon" title="c3">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 <b>by </b><i><a class="txt" href="waybel_2.html#E7:47"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>


</div>
