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<b>let </b><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel19.html#NM1" title="WAYBEL19:NM.1">TopPoset</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">f</font> being <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b3">Y</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b2">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b3">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span><br/><b>let </b><font color="Maroon" title="c3">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">Y</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b2">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Olive" title="b1">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b2">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span><br/><b>let </b><font color="Maroon" title="c4">f</font> be  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>  st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span><br/><b>let </b><font color="Maroon" title="c5">Y</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V2" title="WAYBEL_0:attr.2">filtered</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  )</span><br/>







<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span><br/>

<b>set </b><font color="Maroon" title="c6">N</font> = <font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">M</font> = <font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> ;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E22">A2</font></i>: 
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> #) &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="waybel_9.html#D8" title="WAYBEL_9:def.8">WAYBEL_9:def 8</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E23">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">T</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T27" title="WAYBEL19:th.27">WAYBEL19:27</a>, <a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green" title="E24">A4</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  &amp; the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="ref" href="waybel19.html#T27" title="WAYBEL19:th.27">WAYBEL19:27</a>, <a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<a NAME="E5:39"/><b>then </b><i><font color="Green" title="E25">A5</font></i>: 
( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">T</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>  )
 
<b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></i>;<br/>
<a NAME="E6:39"/><b>then </b><i><font color="Green" title="E26">A6</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c3">X</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> )
 
<b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c3">X</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E5:39"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">g</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_6.html#D12" title="YELLOW_6:def.12">YELLOW_6:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">g</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive">b<sub>3</sub></font> ) ) )</span><br/>

<b>thus </b><a NAME="E8:39"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E5:39"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="#E6:39"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>  ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> st <br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive">b<sub>3</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c9">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> st <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c9">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive">b<sub>2</sub></font> )</span><br/>

<b>consider </b><font color="Maroon" title="c10">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:39"/><i><font color="Green" title="E27">A7</font></i>: 
( <font color="Maroon" title="c10">n</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 <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c10">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> &amp; <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> )
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="funct_2.html#T115" title="FUNCT_2:th.115">FUNCT_2:115</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">n</font> = <font color="Maroon" title="c10">n</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E10:39"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>  holds <br/>(  not <font color="Maroon" title="c11">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c9">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c12">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c11">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> or <font color="Maroon" title="c9">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c12">p</font> )</span><br/>

<a NAME="E12:39"/>
<font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E24">A4</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">n'</font> = <font color="Maroon" title="c11">n</font>, <font color="Maroon" title="c14">p'</font> = <font color="Maroon" title="c12">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="#E10:39"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">fp</font> = <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">p'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:39"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="funct_2.html#T43" title="FUNCT_2:th.43">FUNCT_2:43</a></i>;<br/>
<a NAME="E15:39"/>
<font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> 
 
<b>by </b><i><a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<a NAME="E16:39"/><b>then </b><i><font color="Green" title="E28">A8</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="waybel_9.html#K7" title="WAYBEL_9:func.7">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> is    <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c1">S</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  &amp; <font color="Maroon" title="c13">n'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n'</font> &amp; <font color="Maroon" title="c14">p'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p'</font> )
 
<b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="waybel21.html#T12" target="_self" title="WAYBEL21:th.12">Th12</a></i>;<br/>
<a NAME="E17:39"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a>  is  <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c2">T</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> 
 
<b>by </b><i><a class="ref" href="yellow_9.html#T7" title="YELLOW_9:th.7">YELLOW_9:7</a></i>;<br/>
<b>assume </b><a NAME="E18:39"/>
<font color="Maroon" title="c11">n</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c12">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c12">p</font></span><br/>

<a NAME="E19:39"/><b>then </b>
<font color="Maroon" title="c13">n'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c14">p'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
 
<b>by </b><i><a class="txt" href="#E16:39"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="yellow_0.html#T60" title="YELLOW_0:th.60">YELLOW_0:60</a></i>;<br/>
<a NAME="E20:39"/><b>then </b>
<font color="Maroon" title="c13">n'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c14">p'</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/>
<a NAME="E21:39"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">n'</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">p'</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#D2" title="WAYBEL_1:def.2">WAYBEL_1:def 2</a></i>;<br/>
<a NAME="E22:39"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">n'</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">p'</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">n'</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">m</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">p'</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">fp</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">Y</font> <a href="yellow_9.html#K3" title="YELLOW_9:func.3">opp+id</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E10:39"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/>
<a NAME="E23:39"/><b>then </b>
<font color="Maroon" title="c15">fp</font> <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c9">m</font>
 
<b>by </b><i><a class="txt" href="#E17:39"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="yellow_0.html#T61" title="YELLOW_0:th.61">YELLOW_0:61</a></i>;<br/>
<b>hence </b><a NAME="E24:39"/>
<font color="Maroon" title="c9">m</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c8">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c12">p</font>
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E4:39"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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