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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">p</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> is  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>  holds <br/>for <font color="Olive" title="b2">Lk</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">k</font> where <font color="Olive" title="b3">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">k</font> </span>}</span>  holds <br/>( ( <font color="Olive" title="b1">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b2">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  ) ) &amp; ( <font color="Olive" title="b1">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b2">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</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"> ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</a>  implies for <font color="Olive" title="b1">Lk</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">k</font> </span>}</span>  holds <br/>( ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  ) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:90"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="waybel_1.html#V6" title="WAYBEL_1:attr.6">projection</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">Lk</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">k</font> where <font color="Olive" title="b2">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">k</font> </span>}</span>  holds <br/>( ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  ) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Olive" title="b1">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) ) )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">Lc</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">c</font> where <font color="Olive" title="b1">c</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Olive" title="b1">c</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">c</font> </span>}</span>  as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T46" target="_self" title="WAYBEL_1:th.46">Th46</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">Lk</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">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="c4">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</font> </span>}</span>  implies ( ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  ) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:90"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c4">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">k</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="c2">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  ) ) &amp; ( <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) ) )</span><br/>

<a NAME="E4:90"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">Def13</a></span>;<br/>
<a NAME="E5:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A4</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel_1.html#T52" target="_self" title="WAYBEL_1:th.52">Th52</a></span>;<br/>
<a NAME="E6:90"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c3">Lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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">Lc</font></span>)</span>
 
<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="lab"><font color="Green" title="E62">A6</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_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">p</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/>.= 
<font color="Maroon" title="c3">Lc</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c4">Lk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="waybel_1.html#T45" target="_self" title="WAYBEL_1:th.45">Th45</a></span>
;<br/>
<a NAME="E8:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A7</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_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Lk</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E9:90"/><span class="lab"><font color="Green" title="E64">A8</font></span>: 
<font color="Maroon" title="c4">Lk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 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="c4">Lk</font></span>)</span>
 
<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/>
<a NAME="E10:90"/><span class="lab"><font color="Green" title="E65">A9</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_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Lc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<div><span class="kw">hereby </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="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>  implies (  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:90_2"/><span class="lab"><font color="Green" title="E66">A10</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</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#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  )</span><br/><span class="kw">thus </span><a NAME="E2:90_2"/><span class="lab"><font color="Green" title="E67">A11</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</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_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">Lk</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span> )</span><br/>

<a NAME="E1:90_2_1"/>
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="c4">Lk</font></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">X9</font> = <font color="Maroon" title="c5">X</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:90_2_1"/><span class="lab"><font color="Green" title="E68">A12</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">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">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span></span><br/>

<a NAME="E4:90_2_1"/><span class="lab"><font color="Green" title="E69">A13</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font> or <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_2_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</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">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</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="E3:90_2_1_1"/><span class="lab"><font color="Green" title="E70">A14</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X9</font>
 <span class="kw">and </span><br/><a NAME="E4:90_2_1_1"/><span class="lab"><font color="Green" title="E71">A15</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:90_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Lk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E3:90_2_1_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a></span>;<br/>
<a NAME="E7:90_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A16</font></span>: 
 ex <font color="Olive" title="b1">x9</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">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">x9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/>
<a NAME="E8:90_2_1_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c6">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2_1"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>;<br/>
<a NAME="E9:90_2_1_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2_1_1"><span class="lab"><font color="Green" title="E70">A14</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:90_2_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_2_1_1"><span class="lab"><font color="Green" title="E71">A15</font></span></a>, <a class="txt" href="#E7:90_2_1_1"><span class="lab"><font color="Green" title="E72">A16</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:90_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c6">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_2"><span class="lab"><font color="Green" title="E66">A10</font></span></a>, <a class="ref" href="waybel_0.html#D33" title="WAYBEL_0:def.33">WAYBEL_0:def 33</a></span>;<br/>
<a NAME="E6:90_2_1"/><span class="kw">then </span>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_2_1"><span class="lab"><font color="Green" title="E68">A12</font></span></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></span>;<br/>
<a NAME="E7:90_2_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_2_1"><span class="lab"><font color="Green" title="E69">A13</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:90_2_1"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</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">thus </span><a NAME="E3:90_2"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_2_2"/><span class="lab"><font color="Green" title="E68">A17</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">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">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span></span><br/>

<a NAME="E2:90_2_2"/>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">Lk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E3:90_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A18</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E2:90_2"><span class="lab"><font color="Green" title="E67">A11</font></span></a>, <a class="txt" href="#E1:90_2_2"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a></span>;<br/>
<a NAME="E4:90_2_2"/>
(  <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font> is  <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  &amp; <font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">Lc</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E10:90"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="waybel_1.html#T52" target="_self" title="WAYBEL_1:th.52">Th52</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E5:90_2_2"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:90_2_2"><span class="lab"><font color="Green" title="E68">A17</font></span></a>, <a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:90_2_2"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E3:90_2_2"><span class="lab"><font color="Green" title="E69">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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>

<span class="kw">assume </span><a NAME="E12:90"/><span class="lab"><font color="Green" title="E66">A19</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</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#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  &amp;  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E13:90"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
 <a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</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_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">X</font> be   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">Lk</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_3"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">X9</font> = <font color="Maroon" title="c5">X</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T7" title="YELLOW_2:th.7">YELLOW_2:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:90_3"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">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">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span></span><br/>

<a NAME="E4:90_3"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font> or <font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:90_3_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</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">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</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="E3:90_3_1"/><span class="lab"><font color="Green" title="E70">A23</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X9</font>
 <span class="kw">and </span><br/><a NAME="E4:90_3_1"/><span class="lab"><font color="Green" title="E71">A24</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E6:90_3_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Lk</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E3:90_3_1"><span class="lab"><font color="Green" title="E70">A23</font></span></a></span>;<br/>
<a NAME="E7:90_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A25</font></span>: 
 ex <font color="Olive" title="b1">x9</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">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">x9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>;<br/>
<a NAME="E8:90_3_1"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="lattice3.html#NR4" title="LATTICE3:NR.4">is_&gt;=_than</a> <font color="Maroon" title="c6">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>;<br/>
<a NAME="E9:90_3_1"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3_1"><span class="lab"><font color="Green" title="E70">A23</font></span></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:90_3_1"/>
<font color="Maroon" title="c7">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_3_1"><span class="lab"><font color="Green" title="E71">A24</font></span></a>, <a class="txt" href="#E7:90_3_1"><span class="lab"><font color="Green" title="E72">A25</font></span></a>, <a class="ref" href="orders_2.html#T26" title="ORDERS_2:th.26">ORDERS_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:90_3"/>
<font color="Maroon" title="c2">p</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c6">X9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:90"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></span>;<br/>
<a NAME="E6:90_3"/><span class="kw">then </span>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> &amp;  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c6">X9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90_3"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></span>;<br/>
<a NAME="E7:90_3"/><span class="kw">then </span>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:90_3"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="yellow_0.html#T30" title="YELLOW_0:th.30">YELLOW_0:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:90_3"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:90"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</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">let </span><font color="Maroon" title="c5">X</font> be   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> or  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E14:90"/><span class="lab"><font color="Green" title="E68">A26</font></span>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E15:90"/><span class="lab"><font color="Green" title="E69">A27</font></span>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c5">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">  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span></span><br/>

<a NAME="E16:90"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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="c4">Lk</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:90"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E17:90"/><span class="kw">then </span>
<font color="Maroon" title="c5">X</font> is   <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c4">Lk</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T8" title="YELLOW_2:th.8">YELLOW_2:8</a></span>;<br/>
<a NAME="E18:90"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A28</font></span>: 
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c4">Lk</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:90"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="txt" href="#E14:90"><span class="lab"><font color="Green" title="E68">A26</font></span></a>, <a class="txt" href="#E15:90"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="ref" href="waybel_0.html#D4" title="WAYBEL_0:def.4">WAYBEL_0:def 4</a></span>;<br/>
<a NAME="E19:90"/>
<font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 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">Lc</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:90"><span class="lab"><font color="Green" title="E65">A9</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E20:90"/><span class="kw">then </span>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_0.html#K5" title="YELLOW_0:func.5">subrelstr</a> <font color="Maroon" title="c3">Lc</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:90"><span class="lab"><font color="Green" title="E69">A27</font></span></a>, <a class="txt" href="#E5:90"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="yellow_0.html#D19" title="YELLOW_0:def.19">YELLOW_0:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:90"/>
 <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c5">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:90"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="txt" href="#E6:90"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E9:90"><span class="lab"><font color="Green" title="E64">A8</font></span></a>, <a class="txt" href="#E18:90"><span class="lab"><font color="Green" title="E70">A28</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0: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>
