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<b>let </b><font color="Maroon">C1</font> be  <a href="altcat_1.html#V6">strict</a> <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">C2</font> be  <a href="altcat_1.html#V6">strict</a> <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_2"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">x</font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/> ( <font color="Olive">x</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font> iff ( <font color="Olive">x</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) )
 <b>and </b><br/><a NAME="E2:48_1_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font></span>)</span> holds <br/> ( <font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Olive">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>and </b><br/><a NAME="E3:48_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/> ( <font color="Olive">x</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font> iff ( <font color="Olive">x</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) )
 <b>and </b><br/><a NAME="E4:48_1_2"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font></span>)</span> holds <br/> ( <font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Olive">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">w</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="wellord1.html#V2">well-ordering</a> <a href="yellow21.html#V5" target="_self">upper-bounded</a> <a href="orders_1.html#NM2">Order</a> of <font color="Maroon">w</font>;<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> ] <b>means </b> ex <font color="Olive">L1</font>, <font color="Olive">L2</font> being  <a href="waybel_0.html#NM8">LATTICE</a> ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">L1</font>,<font color="Olive">L2</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">L1</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">L2</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> );<br/>
<div><i><font color="Green">E50</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C1</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font></span>)</span>;<br/><a NAME="E1:48_1_2_1"/>
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> implies <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> ) &amp; ( <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:48_1_2_1"/>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C2</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font></span>)</span>;<br/><a NAME="E1:48_1_2_2"/>
(  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp;  <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> implies <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> ) &amp; ( <font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> ) )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:48_1_2_2"/>
( <font color="Maroon">f</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">a</font>,<font color="Maroon">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">f</font>] )
 ;<br/></div><b>end;</b></div>
<a NAME="E9:48_1_2"/>
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>  st ( for <font color="Olive">x</font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/> ( <font color="Olive">x</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C1</font> iff ( <font color="Olive">x</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) ) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C1</font><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font></span>)</span> holds <br/> ( <font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] ) ) &amp; ( for <font color="Olive">x</font> being  <a href="waybel_0.html#NM8">LATTICE</a> holds <br/> ( <font color="Olive">x</font> is   <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C2</font> iff ( <font color="Olive">x</font> is <a href="orders_2.html#V1">strict</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> ) ) ) &amp; ( for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="altcat_1.html#NM1">object</a> of <font color="Olive">C2</font><br/> for <font color="Olive">f</font> being <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">a</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow21.html#K3" target="_self">latt</a> <font color="Olive">b</font></span>)</span> holds <br/> ( <font color="Olive">f</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">a</font>,<font color="Olive">b</font></span><a href="altcat_1.html#K1">^&gt;</a></span> iff <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] ) ) holds <br/> <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C1</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">C1</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">C1</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">C2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Olive">C2</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Olive">C2</font> #)
 
<b>from </b><i><a class="ref" href="yellow21.html#S5" target="_self">YELLOW21:sch 5</a>();<br/></i>
<b>hence </b><a NAME="E10:48_1_2"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i>, , , </i>;<br/>


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