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<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  <a href="altcat_1.html#V6">strict</a> <a href="yellow21.html#V2">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_1_2"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>and </b><br/><a NAME="E2:25_1_2"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Olive">b<sub>3</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 <b>and </b><br/><a NAME="E3:25_1_2"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/>( <font color="Olive">b<sub>1</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>and </b><br/><a NAME="E4:25_1_2"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>2</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Olive">b<sub>3</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> );<br/>
<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><a NAME="E1:25_1_2_1"/>
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> implies <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> ) &amp; ( <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="waybel34.html#E2:25_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><b>hence </b><a NAME="E2:25_1_2_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><a NAME="E1:25_1_2_2"/>
(  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="yellow21.html#K3">latt</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> implies <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> ) &amp; ( <font color="Maroon">c<sub>7</sub></font> is <a href="waybel_0.html#V18">sups-preserving</a> implies <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="waybel34.html#E4:25_1_2"><i><font color="Green">E19</font></i></a></i>;<br/><b>hence </b><a NAME="E2:25_1_2_2"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/></div><b>end;</b></div>
<a NAME="E7:25_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="yellow21.html#V2">lattice-wise</a> <a href="altcat_1.html#NM4">category</a> st ( for <font color="Olive">b<sub>3</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/>( <font color="Olive">b<sub>3</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">b<sub>1</sub></font> iff ( <font color="Olive">b<sub>3</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) ) &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>4</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>] ) ) &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/>( <font color="Olive">b<sub>3</sub></font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">b<sub>2</sub></font> iff ( <font color="Olive">b<sub>3</sub></font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> ) ) ) &amp; ( for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>5</sub></font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3">latt</a> <font color="Olive">b<sub>4</sub></font></span>)</span> holds <br/>( <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>] ) ) holds <br/> <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">b<sub>1</sub></font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">b<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>2</sub></font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">b<sub>2</sub></font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">b<sub>2</sub></font> #)
 
<b>from </b><i><a class="ref" href="yellow21.html#S5">YELLOW21:sch 5</a>();<br/></i>
<b>hence </b><a NAME="E8:25_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:25_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="waybel34.html#E3:25_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="waybel34.html#E5:25_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="waybel34.html#E6:25_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>


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