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<b>let </b><font color="Maroon">A1</font> be  <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>, <font color="Maroon">A2</font> be  <a href="yellow21.html#V2" target="_self">lattice-wise</a> <a href="altcat_1.html#NM4">category</a>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:21_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">C1</font>, <font color="Olive">C2</font> being <a href="altcat_1.html#V9">semi-functional</a> <a href="yellow18.html#V2">para-functional</a> <a href="altcat_1.html#NM4">category</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></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> holds  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</font>) ) &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></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> holds  <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> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">a</font>,<font color="Olive">b</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="yellow18.html#S19">YELLOW18:sch 19</a>();<br/></i>
<b>assume </b><b>that </b><br/><a NAME="E2:21_1"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E3:21_1"/><i><font color="Green">E40</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">A1</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">P<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] )
 <b>and </b><br/><a NAME="E4:21_1"/><i><font color="Green">E41</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E5:21_1"/><i><font color="Green">E43</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">A2</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">P<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">f</font>] )
 ;<br/>

<a NAME="E6:21_1"/><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">A1</font> holds  <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> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">a</font>,<font color="Olive">b</font>
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_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">A2</font>, <font color="Maroon">b</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A2</font>;<br/><b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">A1</font> <b>by </b><i>, </i>;<br/><a NAME="E2:21_1_1"/><i><font color="Green">E48</font></i>: 
<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> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>
 ;<br/><b>thus </b><a NAME="E3:21_1_1"/>
<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> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_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">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:21_1_1_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</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>
 ;<br/><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a</font>,<font color="Maroon">b</font> ;<br/><a NAME="E3:21_1_1_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:21_1_1_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a'</font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b'</font>, <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font>] &amp;  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> is   <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> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:21_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:21_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">a'</font>,<font color="Maroon">b'</font> ;<br/>
<a NAME="E4:21_1_1_1"/>
 <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:21_1_1_1"/><b>then </b>
( <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">a</font>, <a href="yellow21.html#K3" target="_self">latt</a> <font color="Maroon">b</font>, <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font>] &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="yellow21.html#K5" target="_self">@</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:21_1_1_1"/>
<font color="Maroon">x</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/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E8:21_1"/>
 <a href="altcat_1.html#G2">AltCatStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A1</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A1</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A1</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="Maroon">A2</font>,the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">A2</font>,the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">A2</font> #)
 <b>by </b><i>, , , </i>;<br/>


</div>
