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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c3">D1</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>;<br/><b>let </b><font color="Maroon" title="c4">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>;<br/>





<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E24">A1</font></i>: 
(  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c1">S1</font> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c4">D2</font>,<font color="Maroon" title="c2">S2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">s</font> =  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>;<br/>
<a NAME="E2:77"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" target="_self" title="YELLOW_3:def.2">Def2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">s1</font>, <font color="Maroon" title="c7">s2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:77"/><i><font color="Green" title="E25">A2</font></i>: 
( <font color="Maroon" title="c6">s1</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">S1</font> &amp; <font color="Maroon" title="c7">s2</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="c2">S2</font> &amp;  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> <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="c6">s1</font>,<font color="Maroon" title="c7">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><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="c8">s1</font> = <font color="Maroon" title="c6">s1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="yellow_3.html#E4:77"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">s2</font> = <font color="Maroon" title="c7">s2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="txt" href="yellow_3.html#E4:77"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/>
<a NAME="E7:77"/><i><font color="Green" title="E26">A3</font></i>: 
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:77"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow_3.html#T40" target="_self" title="YELLOW_3:th.40">Th40</a></i>;<br/>
<a NAME="E8:77"/><b>then </b><i><font color="Green" title="E27">A4</font></i>: 
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">s2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E4:77"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E9:77"/><b>then </b><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">D1</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#T32" target="_self" title="YELLOW_3:th.32">Th32</a></i>;<br/>
<a NAME="E10:77"/><i><font color="Green" title="E29">A6</font></i>: 
for <font color="Olive" title="b1">b</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">S1</font>,<font color="Maroon" title="c2">S2</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> holds <br/><span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c8">s1</font>,<font color="Maroon" title="c9">s2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E4:77"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="yellow_3.html#E7:77"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E11:77"/><b>then </b>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">D1</font> holds <br/><font color="Maroon" title="c8">s1</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:77"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="yellow_3.html#T36" target="_self" title="YELLOW_3:th.36">Th36</a></i>;<br/>
<a NAME="E12:77"/><b>then </b><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c3">D1</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E9:77"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>
<a NAME="E13:77"/><i><font color="Green" title="E31">A8</font></i>: 
<font color="Maroon" title="c9">s2</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">D2</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E8:77"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="yellow_3.html#T32" target="_self" title="YELLOW_3:th.32">Th32</a></i>;<br/>
<a NAME="E14:77"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st <font color="Olive" title="b1">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">D2</font> holds <br/><font color="Maroon" title="c9">s2</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">b</font>
 
<b>by </b><i><a class="txt" href="yellow_3.html#E1:77"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="yellow_3.html#E10:77"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="yellow_3.html#T36" target="_self" title="YELLOW_3:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E15:77"/>
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <span class="p1"><a href="yellow_3.html#K6" title="YELLOW_3:func.6">[:</a><span class="default"><font color="Maroon" title="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c3">D1</font></span>)</span>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c4">D2</font></span>)</span></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <b>by </b><i><a class="txt" href="yellow_3.html#E4:77"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="yellow_3.html#E12:77"><i><font color="Green" title="E30">A7</font></i></a>, <a class="txt" href="yellow_3.html#E13:77"><i><font color="Green" title="E31">A8</font></i></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></i>;<br/>


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