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<b>let </b><font color="Maroon" title="c3">C1</font>, <font color="Maroon" title="c4">C2</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:24_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">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="c3">C1</font>
 <b>and </b><br/><a NAME="E2:24_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C1</font>
 <b>and </b><br/><a NAME="E3:24_1_1"/><i><font color="Green" title="E8">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C1</font><br/> for <font color="Olive" title="b4">a'</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">c'</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C2</font>  st <font color="Olive" title="b4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b5">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> holds <br/>the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b5">b'</font>,<font color="Olive" title="b6">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E4:24_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">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="c4">C2</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:24_1_1"><i><font color="Green" title="E6">A1</font></i></a></i>;<br/>

<a NAME="E5:24_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C1</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:24_1_1"/>
the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_2.html#K1" title="ALTCAT_2:func.1">~</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:24_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_4.html#T53" title="FUNCT_4:th.53">FUNCT_4:53</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c4">C2</font>;<br/><b>let </b><font color="Maroon" title="c8">a'</font>, <font color="Maroon" title="c9">b'</font>, <font color="Maroon" title="c10">c'</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C1</font>;<br/>



<b>assume </b><a NAME="E7:24_1_1"/>
( <font color="Maroon" title="c8">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c9">b'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 ;<br/>

<a NAME="E8:24_1_1"/><b>then </b><i><font color="Green" title="E9">A4</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font></span>)</span>
 
<b>by </b><i><a class="txt" href="yellow18.html#E3:24_1_1"><i><font color="Green" title="E8">A3</font></i></a></i>;<br/>
<a NAME="E9:24_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font></span>)</span>,<span class="p2">(<span class="default">the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c3">C1</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<b>hence </b><a NAME="E10:24_1_1"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c3">C1</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c8">a'</font>,<font color="Maroon" title="c9">b'</font>,<font color="Maroon" title="c10">c'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor0.html#K1" title="FUNCTOR0:func.1">~</a> <span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c4">C2</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E8:24_1_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="funct_4.html#T53" title="FUNCT_4:th.53">FUNCT_4:53</a></i>;<br/>


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