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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#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Olive" title="b3">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Olive" title="b1">D1</font> &amp; <font color="Olive" title="b4">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Olive" title="b2">D2</font> 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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Olive" title="b2">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b3">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Olive" title="b1">D2</font> 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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Olive" title="b1">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b2">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D2</font> 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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c5">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Olive" title="b1">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D2</font> 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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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></span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c6">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D2</font> implies <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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c3">D1</font>
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_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></span><br/>

<b>let </b><font color="Maroon" title="c7">b</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c7">b</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="c3">D1</font>,<font color="Maroon" title="c4">D2</font></span><a href="yellow_3.html#K6" title="YELLOW_3:func.6">:]</a></span> or <font color="Maroon" title="c7">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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> )</span><br/>

<b>assume </b><a NAME="E3:60"/>
<font color="Maroon" title="c7">b</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="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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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></span><br/>

<b>then consider </b><font color="Maroon" title="c8">b1</font>, <font color="Maroon" title="c9">b2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:60"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D1</font> &amp; <font color="Maroon" title="c9">b2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">D2</font> &amp; <font color="Maroon" title="c7">b</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">b1</font>,<font color="Maroon" title="c9">b2</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="c10">b1</font> = <font color="Maroon" title="c8">b1</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="#E5:60"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">b2</font> = <font color="Maroon" title="c9">b2</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="#E5:60"><i><font color="Green" title="E14">A3</font></i></a></i>;<br/>
<a NAME="E8:60"/>
( <font color="Maroon" title="c10">b1</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )
 
<b>by </b><i><a class="txt" href="#E1:60"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E2:60"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="#E5:60"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:60"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <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>
 <b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="yellow_3.html#T11" target="_self" title="YELLOW_3:th.11">Th11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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