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<a NAME="E1:19_1"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b5">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">f</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b4">f</font>]
 
<b>by </b><i><a class="txt" href="#E2:19"><i><font color="Green" title="E10">A2</font></i></a></i>;<br/>
<a NAME="E2:19_1"/><i><font color="Green" title="E13">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>, <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Olive" title="b1">a</font>]
 
<b>by </b><i><a class="txt" href="#E3:19"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c1">C</font> being  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a><b> such that </b><br/><a NAME="E4:19_1"/><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c1">C</font> is <a href="yellow21.html#V2" title="YELLOW21:attr.2">lattice-wise</a>
 <b>and </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E15">A7</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E6:19_1"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a><br/> for <font color="Olive" title="b3">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="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) )
 <b>from </b><i><a class="ref" href="yellow21.html#S1" target="_self" title="YELLOW21:sch.1">YELLOW21:sch 1</a>(<a class="txt" href="#E1:19"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E1:19_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="#E2:19_1"><i><font color="Green" title="E13">A5</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c2">C</font> = <font color="Maroon" title="c1">C</font> as  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="yellow21.html#V2" title="YELLOW21:attr.2">lattice-wise</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a> <b>by </b><i><a class="txt" href="#E4:19_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c2">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b1">a</font>, <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) ) )</span><br/>

<b>thus </b><a NAME="E8:19_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="#E5:19_1"><i><font color="Green" title="E15">A7</font></i></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" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b3">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 <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b1">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b1">a</font>, <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] )</span><br/>

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">C</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 <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c4">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c3">a</font>, <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">f</font>] )</span><br/>

<b>thus </b><a NAME="E9:19_1"/>
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 <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c3">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c4">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c3">a</font>, <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c4">b</font>,<font color="Olive" title="b1">f</font>] )
 <b>by </b><i><a class="txt" href="#E5:19_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="#E6:19_1"><i><font color="Green" title="E16">A8</font></i></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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