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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="lattice3.html#V3" title="LATTICE3:attr.3">complete</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/> <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Olive" title="b1">p</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</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  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  )</span><br/>



<a NAME="E1:92"/><span class="lab"><font color="Green" title="E58">A1</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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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/>
<span class="kw">assume </span><a NAME="E2:92"/><span class="lab"><font color="Green" title="E59">A2</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">  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</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> , <font color="Maroon" title="c4">Lk</font> = <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>  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/>
<a NAME="E4:92"/><span class="lab"><font color="Green" title="E60">A3</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">Lc</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Lc</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="#E2:92"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="waybel_1.html#T45" target="_self" title="WAYBEL_1:th.45">Th45</a>, <a class="ref" href="yellow_0.html#D15" title="YELLOW_0:def.15">YELLOW_0:def 15</a></span>;<br/>
<a NAME="E5:92"/>
<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="#E2:92"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="waybel_1.html#D13" target="_self" title="WAYBEL_1:def.13">Def13</a></span>;<br/>
<a NAME="E6:92"/><span class="kw">then </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="E7:92"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">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="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_2.html#T33" title="YELLOW_2:th.33">YELLOW_2:33</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">pc</font> = <font color="Maroon" title="c2">p</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">Lc</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="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="#E2:92"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="waybel_1.html#T48" target="_self" title="WAYBEL_1:th.48">Th48</a></span>;<br/>
<a NAME="E9:92"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c5">pc</font> is  <a href="yellow_0.html#V7" title="YELLOW_0:attr.7">infs-inheriting</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:92"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="ref" href="waybel_1.html#T50" target="_self" title="WAYBEL_1:th.50">Th50</a>, <a class="ref" href="waybel_1.html#T56" target="_self" title="WAYBEL_1:th.56">Th56</a></span>;<br/>
<span class="lab"><font color="Green" title="E63">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="c5">pc</font></span>)</span> = 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">pc</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/>.= 
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="#E2:92"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E1:92"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="ref" href="waybel_1.html#T47" target="_self" title="WAYBEL_1:th.47">Th47</a></span>
;<br/>
<span class="kw">then </span>the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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="c5">pc</font></span>)</span> = 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</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="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default">the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<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>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</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="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></span>
<br/>.= 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</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="#E1:92"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E4:92"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="wellord1.html#T29" title="WELLORD1:th.29">WELLORD1:29</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>
<br/>.= 
the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</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="ref" href="yellow_0.html#D14" title="YELLOW_0:def.14">YELLOW_0:def 14</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:92"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">p</font> is  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:92"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="txt" href="#E9:92"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="txt" href="#E10:92"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="yellow_2.html#T32" title="YELLOW_2:th.32">YELLOW_2:32</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/>


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