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<b>let </b><font color="Maroon" title="c1">C</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> <a href="altcat_1.html#V11" title="ALTCAT_1:attr.11">associative</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font>, <font color="Maroon" title="c4">o3</font>, <font color="Maroon" title="c5">o4</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:53"/><i><font color="Green" title="E34">A1</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:53"/><i><font color="Green" title="E35">A2</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E3:53"/><i><font color="Green" title="E36">A3</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>let </b><font color="Maroon" title="c6">a</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>;<br/><b>let </b><font color="Maroon" title="c7">b</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>;<br/><b>let </b><font color="Maroon" title="c8">c</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font>;<br/>





<a NAME="E4:53"/><i><font color="Green" title="E37">A4</font></i>: 
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> is <a href="altcat_1.html#V3" title="ALTCAT_1:attr.3">associative</a>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D17" target="_self" title="ALTCAT_1:def.17">Def17</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E38">A5</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:53"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="altcat_1.html#E2:53"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></i>;<br/>
<a NAME="E6:53"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:53"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="altcat_1.html#E2:53"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green" title="E40">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c5">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:53"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:53"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="altcat_1.html#D4" target="_self" title="ALTCAT_1:def.4">Def4</a></i>;<br/>
<a NAME="E8:53"/><i><font color="Green" title="E41">A8</font></i>: 
<font color="Maroon" title="c8">c</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="altcat_1.html#E2:53"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:53"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>;<br/>
<a NAME="E9:53"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="altcat_1.html#E1:53"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="altcat_1.html#E2:53"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="altcat_1.html#E3:53"><i><font color="Green" title="E36">A3</font></i></a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font> &amp; <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font> )
 
;<br/>
<b>thus </b><font color="Maroon" title="c8">c</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">b</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">a</font></span>)</span> = 
<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="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">c</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">a</font></span>)</span>
<b>by </b><i><a class="txt" href="altcat_1.html#E3:53"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="altcat_1.html#E5:53"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="altcat_1.html#E6:53"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<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="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c5">o4</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c4">o3</font>,<font color="Maroon" title="c5">o4</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font></span>)</span>,<font color="Maroon" title="c6">a</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E4:53"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="altcat_1.html#E10:53"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="altcat_1.html#D7" target="_self" title="ALTCAT_1:def.7">Def7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">c</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">a</font>
<b>by </b><i><a class="txt" href="altcat_1.html#E1:53"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="altcat_1.html#E7:53"><i><font color="Green" title="E40">A7</font></i></a>, <a class="txt" href="altcat_1.html#E8:53"><i><font color="Green" title="E41">A8</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
;<br/>


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