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<b>let </b><font color="Maroon" title="c1">A1</font>, <font color="Maroon" title="c2">A2</font> be  <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>; <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  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A1</font> iff ( <font color="Olive" title="b1">x</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">A1</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>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) ) &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A2</font> iff ( <font color="Olive" title="b1">x</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">A2</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>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) ) implies  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">A1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">A2</font> #) )</span><br/>

<a NAME="E1:22_1"/><i><font color="Green" title="E9">A1</font></i>: 
for <font color="Olive" title="b1">C1</font>, <font color="Olive" title="b2">C2</font> being <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>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C1</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 <font color="Maroon" title="c1">A1</font> &amp; ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b1">C1</font><br/> for <font color="Olive" title="b5">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="b3">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="b4">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b5">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="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">f</font>] ) ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C2</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 <font color="Maroon" title="c1">A1</font> &amp; ( for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Olive" title="b2">C2</font><br/> for <font color="Olive" title="b5">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="b3">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="b4">b</font></span>)</span> holds <br/> ( <font color="Olive" title="b5">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="b3">a</font>,<font color="Olive" title="b4">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> iff <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">f</font>] ) ) holds <br/> <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">C1</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Olive" title="b1">C1</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Olive" title="b1">C1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">C2</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Olive" title="b2">C2</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Olive" title="b2">C2</font> #)
 
<b>from </b><i><a class="ref" href="yellow21.html#S4" target="_self" title="YELLOW21:sch.4">YELLOW21:sch 4</a>();<br/></i>
<b>assume </b><b>that </b><br/><a NAME="E2:22_1"/><i><font color="Green" title="E10">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A1</font> iff ( <font color="Olive" title="b1">x</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) )
 <b>and </b><br/><a NAME="E3:22_1"/><i><font color="Green" title="E11">A3</font></i>: 
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="c1">A1</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>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] )
 <b>and </b><br/><a NAME="E4:22_1"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> holds <br/> ( <font color="Olive" title="b1">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A2</font> iff ( <font color="Olive" title="b1">x</font> is <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <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">A2</font> ex <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> st <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> implies <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) implies ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] &amp; not <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> ) ) or  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">A1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">A2</font> #) )</span><br/>

<a NAME="E5:22_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A2</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 <font color="Maroon" title="c1">A1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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="c1">A1</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 <font color="Maroon" title="c2">A2</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</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="c2">A2</font> implies <font color="Maroon" title="c3">x</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">A1</font> )</span><br/><b>assume </b><a NAME="E1:22_1_1_1"/><i><font color="Green" title="E13">A5</font></i>: 
<font color="Maroon" title="c3">x</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="c2">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</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">A1</font></span><br/><a NAME="E2:22_1_1_1"/><b>then </b><i><font color="Green" title="E14">A6</font></i>: 
<font color="Maroon" title="c3">x</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 <b>by </b><i><a class="ref" href="yellow21.html#D4" target="_self" title="YELLOW21:def.4">Def4</a></i>;<br/><a NAME="E3:22_1_1_1"/><b>then </b><i><font color="Green" title="E15">A7</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a></i>;<br/><a NAME="E4:22_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</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="c3">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 <b>by </b><i><a class="txt" href="yellow21.html#E4:22_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="yellow21.html#E1:22_1_1_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="yellow21.html#E2:22_1_1_1"><i><font color="Green" title="E14">A6</font></i></a></i>;<br/><a NAME="E5:22_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A1</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E2:22_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="yellow21.html#E3:22_1_1_1"><i><font color="Green" title="E15">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E6:22_1_1_1"/>
<font color="Maroon" title="c3">x</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">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c3">x</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">A1</font> or <font color="Maroon" title="c3">x</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="c2">A2</font> )</span><br/>

<b>assume </b><a NAME="E2:22_1_1"/><i><font color="Green" title="E13">A8</font></i>: 
<font color="Maroon" title="c3">x</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">A1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</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="c2">A2</font></span><br/>

<a NAME="E3:22_1_1"/><b>then </b><i><font color="Green" title="E14">A9</font></i>: 
<font color="Maroon" title="c3">x</font> is   <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="ref" href="yellow21.html#D4" target="_self" title="YELLOW21:def.4">Def4</a></i>;<br/>
<a NAME="E4:22_1_1"/><b>then </b><i><font color="Green" title="E15">A10</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="yellow21.html#D1" target="_self" title="YELLOW21:def.1">Def1</a></i>;<br/>
<a NAME="E5:22_1_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> is  <a href="orders_2.html#V1" title="ORDERS_2:attr.1">strict</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c3">x</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="c3">x</font> <a href="yellow21.html#K1" title="YELLOW21:func.1">as_1-sorted</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="yellow21.html#E2:22_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="yellow21.html#E2:22_1_1"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="yellow21.html#E3:22_1_1"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>
<a NAME="E6:22_1_1"/><b>then </b>
<font color="Maroon" title="c3">x</font> is   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A2</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:22_1"><i><font color="Green" title="E12">A4</font></i></a>, <a class="txt" href="yellow21.html#E4:22_1_1"><i><font color="Green" title="E15">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:22_1_1"/>
<font color="Maroon" title="c3">x</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="c2">A2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_1"/>
(  ex <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">A2</font> ex <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> st <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> implies <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] ) implies ( <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>] &amp; not <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> ) ) or  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">A1</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">A1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c2">A2</font>,the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c2">A2</font> #) )
 <b>by </b><i><a class="txt" href="yellow21.html#E1:22_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="yellow21.html#E3:22_1"><i><font color="Green" title="E11">A3</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/>


</div>
