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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</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">R</font> being  <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a>   <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a>  <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">C</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Olive" title="b1">R</font><br/> for <font color="Olive" title="b3">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b2">C</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b2">C</font> is  <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a>  &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> holds <br/> ex <font color="Olive" title="b4">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b4">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b4">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b3">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b4">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">R</font> &amp;  ex <font color="Olive" title="b5">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b2">C</font></span>)</span> st <br/>( <font color="Olive" title="b4">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">cs1</font> &amp; <font color="Olive" title="b3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b5">cs1</font> &amp; ( for <font color="Olive" title="b6">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b2">C</font></span>)</span>  st <font color="Olive" title="b3">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b6">a</font> holds <br/><font color="Olive" title="b5">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b6">a</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">R</font> be   <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a>   <a href="waybel_4.html#V2" title="WAYBEL_4:attr.2">auxiliary(ii)</a>  <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</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 non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Olive" title="b1">C</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">C</font> is  <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a>  &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> holds <br/> ex <font color="Olive" title="b3">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b3">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b3">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b4">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">C</font></span>)</span> st <br/>( <font color="Olive" title="b3">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">cs1</font> &amp; <font color="Olive" title="b2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b4">cs1</font> &amp; ( for <font color="Olive" title="b5">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">C</font></span>)</span>  st <font color="Olive" title="b2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b5">a</font> holds <br/><font color="Olive" title="b4">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b5">a</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">C</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</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="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">C</font>  st  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">C</font> is  <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a>  &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> holds <br/> ex <font color="Olive" title="b2">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b2">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b3">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Olive" title="b2">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">cs1</font> &amp; <font color="Olive" title="b1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">cs1</font> &amp; ( for <font color="Olive" title="b4">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Olive" title="b1">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b4">a</font> holds <br/><font color="Olive" title="b3">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b4">a</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">C</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#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">C</font> is  <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a>  &amp; not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> implies  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b2">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">cs1</font> &amp; ( for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b2">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">a</font> ) ) ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">and </span><br/><a NAME="E2:31"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
<font color="Maroon" title="c3">C</font> is  <a href="waybel35.html#V2" title="WAYBEL35:attr.2">maximal</a> 
 ; <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#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> or  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b2">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">cs1</font> &amp; ( for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b2">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">a</font> ) ) ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">s</font> =  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:31"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
not  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b2">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">cs1</font> &amp; ( for <font color="Olive" title="b3">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b3">a</font> holds <br/><font color="Olive" title="b2">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">a</font> ) ) )</span><br/>

<a NAME="E5:31"/><span class="kw">then </span>
not <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="lattice7.html#R1" title="LATTICE7:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T45" title="ZFMISC_1:th.45">ZFMISC_1:45</a></span>;<br/>
<a NAME="E6:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is  not    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="ref" href="waybel35.html#D4" target="_self" title="WAYBEL35:def.4">Def4</a></span>;<br/>
<a NAME="E7:31"/>
 ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:31_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">a</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> holds <br/> ex <font color="Olive" title="b2">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b2">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b2">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b2">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> implies  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E2:31_1_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E20">A9</font></span>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">a</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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:31_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a></span>; <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#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/>

<a NAME="E5:31_1_1"/>
<font color="Maroon" title="c6">a</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c4">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1_1"/><span class="lab"><font color="Green" title="E21">A10</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="txt" href="#E1:31_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font></span><br/><div class="add"><a NAME="E2:31_1_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 ;<br/><a NAME="E3:31_1_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E1:31_1_1_1"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:31_1_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="txt" href="#E2:31_1_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="txt" href="#E1:31_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="waybel_4.html#D5" title="WAYBEL_4:def.5">WAYBEL_4:def 5</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_1_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">a</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_1_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E6:31_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:31_1_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:31"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font></span><br/>

<span class="kw">thus </span><a NAME="E8:31_1_1"/>
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E20">A9</font></span></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>
<span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:31_1"/><span class="lab"><font color="Green" title="E18">A13</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E4:31_1"/><span class="lab"><font color="Green" title="E19">A14</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">and </span><br/><a NAME="E5:31_1"/><span class="lab"><font color="Green" title="E20">A15</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">and </span><br/><a NAME="E6:31_1"/><span class="lab"><font color="Green" title="E21">A16</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">and </span><br/><a NAME="E7:31_1"/><span class="lab"><font color="Green" title="E22">A17</font></span>: 
not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:31_1_2"/>
( ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> ) or ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> ) or ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> ) or ( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E18">A13</font></span></a>, <a class="txt" href="#E4:31_1"><span class="lab"><font color="Green" title="E19">A14</font></span></a>, <a class="txt" href="waybel35.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_1"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_1"/>
 ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E6:31_1"><span class="lab"><font color="Green" title="E21">A16</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_2"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_2"/>
 ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E17">A6</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_3"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_3"/>
 ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E20">A15</font></span></a>, <a class="txt" href="#E7:31_1"><span class="lab"><font color="Green" title="E22">A17</font></span></a>, <a class="txt" href="#E1:31_1"><span class="lab"><font color="Green" title="E17">A6</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_1_2_4"><span class="kw">suppose </span></a><a NAME="E1:31_1_2_4"/>
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:31_1_2_4"/>
 ex <font color="Olive" title="b1">cs</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:31_1"><span class="lab"><font color="Green" title="E21">A16</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">cs</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><span class="kw"> such that </span><br/><a NAME="E9:31"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
<font color="Maroon" title="c6">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E10:31"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">cs</font>
 <span class="kw">and </span><br/><a NAME="E11:31"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">cs</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">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp;  ex <font color="Olive" title="b1">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Maroon" title="c6">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">cs1</font> &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Olive" title="b1">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">a</font> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E12:31"/>
( <font color="Maroon" title="c6">cs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">cs</font> &amp; not <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c6">cs</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E19">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">cs1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> st <br/>( <font color="Maroon" title="c6">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">cs1</font> &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Olive" title="b1">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">a</font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">cs1</font> = <font color="Maroon" title="c6">cs</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:31"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">cs1</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">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">cs1</font> &amp; <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">cs1</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c7">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E14:31"/>
<font color="Maroon" title="c6">cs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">cs1</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="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">cs1</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c7">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font> ) )</span><br/>

<a NAME="E15:31"/><span class="lab"><font color="Green" title="E20">A21</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">cs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:31"><span class="lab"><font color="Green" title="E18">A19</font></span></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E16:31"/>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">cs1</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>  st <font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c7">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> or <font color="Maroon" title="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">cs1</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">b0</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:31_2"/>
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</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="c8">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c7">cs1</font></span><br/>

<a NAME="E3:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="yellow_4.html#T1" title="YELLOW_4:th.1">YELLOW_4:1</a></span>;<br/>
<a NAME="E4:31_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">b0</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">cs</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:31"><span class="lab"><font color="Green" title="E20">A21</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_2"/>
<font color="Maroon" title="c8">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">cs1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</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>

<span class="kw">let </span><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span>; <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="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">a</font> implies <font color="Maroon" title="c7">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">a0</font> = <font color="Maroon" title="c8">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T59" title="YELLOW_0:th.59">YELLOW_0:59</a></span>;<br/>
<a NAME="E18:31"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<span class="kw">assume </span><a NAME="E19:31"/>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c8">a</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">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font></span><br/>

<a NAME="E20:31"/><span class="kw">then </span>
<font color="Maroon" title="c4">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c9">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="ref" href="yellow_0.html#T63" title="YELLOW_0:th.63">YELLOW_0:63</a></span>;<br/>
<a NAME="E21:31"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A23</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c4">X</font>,<font color="Maroon" title="c1">L</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></span>;<br/>
<a NAME="E22:31"/><span class="lab"><font color="Green" title="E23">A24</font></span>: 
<font color="Maroon" title="c6">cs</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">cs</font>
 
;<br/>
<a NAME="E23:31"/>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">cs1</font>,<font color="Maroon" title="c8">a</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">cs1</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c7">cs1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:31"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></span>;<br/>
<a NAME="E24:31"/><span class="kw">then </span>
<font color="Maroon" title="c6">cs</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c9">a0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:31"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="txt" href="#E21:31"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="txt" href="#E22:31"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a>, <a class="ref" href="waybel_4.html#D5" title="WAYBEL_4:def.5">WAYBEL_4:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:31"/>
<font color="Maroon" title="c7">cs1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</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>
