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<span class="kw">let </span><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">D2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font><br/> for <font color="Olive" title="b3">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b4">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Olive" title="b1">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="Olive" title="b2">D2</font>,<font color="Maroon" title="c2">S2</font> &amp; ( for <font color="Olive" title="b5">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="b5">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="Olive" title="b1">D1</font>,<font color="Olive" title="b2">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="Olive" title="b3">x</font>,<font color="Olive" title="b4">y</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="b5">b</font> ) holds <br/>( ( for <font color="Olive" title="b5">c</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="b5">c</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b1">D1</font> holds <br/><font color="Olive" title="b3">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b5">c</font> ) &amp; ( for <font color="Olive" title="b5">d</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="b5">d</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b2">D2</font> holds <br/><font color="Olive" title="b4">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b5">d</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">D1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">D2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b3">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <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="Olive" title="b1">D2</font>,<font color="Maroon" title="c2">S2</font> &amp; ( for <font color="Olive" title="b4">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="b4">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="Olive" title="b1">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="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</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="b4">b</font> ) holds <br/>( ( for <font color="Olive" title="b4">c</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="b4">c</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="Olive" title="b2">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">c</font> ) &amp; ( for <font color="Olive" title="b4">d</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="b4">d</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Olive" title="b1">D2</font> holds <br/><font color="Olive" title="b3">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b4">d</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">D2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font><br/> for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <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> &amp; ( for <font color="Olive" title="b3">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="b3">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="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</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="b3">b</font> ) holds <br/>( ( for <font color="Olive" title="b3">c</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="b3">c</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="Olive" title="b1">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b3">c</font> ) &amp; ( for <font color="Olive" title="b3">d</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="b3">d</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="Olive" title="b2">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b3">d</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>  st  <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> &amp; ( for <font color="Olive" title="b2">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="b2">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="c5">x</font>,<font color="Olive" title="b1">y</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="b2">b</font> ) holds <br/>( ( for <font color="Olive" title="b2">c</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="b2">c</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="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">c</font> ) &amp; ( for <font color="Olive" title="b2">d</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="b2">d</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="Olive" title="b1">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b2">d</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> &amp; ( 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="c5">x</font>,<font color="Maroon" title="c6">y</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> ) implies ( ( for <font color="Olive" title="b1">c</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">c</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="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) &amp; ( for <font color="Olive" title="b1">d</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">d</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="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:66"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:66"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E3:66"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
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="c5">x</font>,<font color="Maroon" title="c6">y</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">c</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">c</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="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font> ) &amp; ( for <font color="Olive" title="b1">d</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">d</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="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:66"/>
for <font color="Olive" title="b1">c</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">c</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="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">c</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">d</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">d</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="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Olive" title="b1">d</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:66_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c4">D2</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">D2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">D1</font> implies <font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:66_1"/>
<font color="Maroon" title="c7">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">D1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E3:66_1"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</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> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E1:66_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="ref" href="yellow_3.html#T33" target="_self" title="YELLOW_3:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:66_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E5:66"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="yellow_0.html#NK4" title="YELLOW_0:NK.4">inf</a> <font color="Maroon" title="c3">D1</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c3">D1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="yellow_0.html#T31" title="YELLOW_0:th.31">YELLOW_0:31</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">D2</font> implies <font color="Maroon" title="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:66"/>
<font color="Maroon" title="c7">b</font> <a href="lattice3.html#R1" title="LATTICE3:pred.1">is_&lt;=_than</a> <font color="Maroon" title="c4">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E7:66"/><span class="kw">then </span>
<span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</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> <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>,<font color="Maroon" title="c7">b</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E5:66"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="yellow_3.html#T33" target="_self" title="YELLOW_3:th.33">Th33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:66"/>
<font color="Maroon" title="c6">y</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
