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<b>let </b><font color="Maroon">C1</font> be  <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a> , <font color="Maroon">C2</font> be  <a href="altcat_1.html#V6">strict</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:37_1"/><i><font color="Green">E20</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>) &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">j</font>,<font color="Olive">k</font>) ) ) ) )
 <b>and </b><br/><a NAME="E2:37_1"/><i><font color="Green">E21</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>) &amp; ( for <font color="Olive">i</font>, <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Olive">i</font>,<font color="Olive">j</font>,<font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">i</font>,<font color="Olive">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Olive">j</font>,<font color="Olive">k</font>) ) ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>thus </b>the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">j</font>)
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>
<br/>.= 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="orders_3.html#E1:37_1"><i><font color="Green">E20</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:37_1"/><b>then </b><i><font color="Green">E30</font></i>: 
the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> the <a href="altcat_1.html#U1">Arrows</a> of <font color="Maroon">C2</font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a>, <a class="txt" href="orders_3.html#E1:37_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="altcat_1.html#T9">ALTCAT_1:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">j</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">k</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() )
 ;<br/><b>hence </b>the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C1</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font> = 
 <a href="altcat_1.html#K6">FuncComp</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">i</font>,<font color="Maroon">j</font>),<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">j</font>,<font color="Maroon">k</font>)
<b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a></i>
<br/>.= 
the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">C2</font> <a href="multop_1.html#K1">.</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>,<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="orders_3.html#E1:37_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#D2" target="_self">Def2</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_1"/>
<font color="Maroon">C1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="txt" href="orders_3.html#E3"><i><font color="Green">Lemma19</font></i></a>, <a class="txt" href="orders_3.html#E1:37_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="orders_3.html#E2:37_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="altcat_1.html#T10">ALTCAT_1:10</a></i>;<br/>


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