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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_1.html#V2">transitive</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="yellow18.html#R2" target="_self">are_opposite</a> 
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="altcat_1.html#V11">associative</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> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default">the <a href="altcat_1.html#U2">Comp</a> of <font color="Maroon">c<sub>1</sub></font> <a href="multop_1.html#K1">.</a> <font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a<sub>4</sub></font>,<font color="Maroon">a<sub>5</sub></font>;<br/>
<div><i><font color="Green">E15</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E16</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E3:30_1"/><i><font color="Green">E17</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self">Th7</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>9</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T7" target="_self">Th7</a></i>;<br/><b>thus </b><font color="Maroon">c<sub>7</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>6</sub></font> = 
<font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>12</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow18.html#E1:30_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E1:30_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="yellow18.html#E3:30_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#D3" target="_self">Def3</a></i>;<br/><b>assume </b><a NAME="E2:30_2"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E3:30_2"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></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">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow18.html#E2:30_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><b>assume </b><a NAME="E5:30_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E6:30_2"/><b>then </b><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow18.html#E5:30_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><b>assume </b><a NAME="E8:30_2"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 ;<br/><a NAME="E9:30_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>9</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>12</sub></font> <b>by </b><i><a class="txt" href="yellow18.html#E1:30"><i><font color="Green">E13</font></i></a>, <a class="txt" href="yellow18.html#E8:30_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="yellow18.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E11:30_2"/><i><font color="Green">E23</font></i>: 
( <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:30_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow18.html#E6:30_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow18.html#E9:30_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="altcat_1.html#D4">ALTCAT_1:def 4</a></i>;<br/><b>thus </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>),<font color="Maroon">c<sub>9</sub></font>) = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>9</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E3:30_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow18.html#E6:30_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="yellow18.html#E9:30_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="yellow18.html#E11:30_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">c<sub>14</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E2:30"><i><font color="Green">E14</font></i></a>, <a class="txt" href="yellow18.html#E3:30_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow18.html#E6:30_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow18.html#E9:30_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="altcat_1.html#T25">ALTCAT_1:25</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">c<sub>16</sub></font>)
<b>by </b><i><a class="txt" href="yellow18.html#E3:30_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="yellow18.html#E11:30_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font>))
<b>by </b><i><a class="txt" href="yellow18.html#E6:30_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="yellow18.html#E9:30_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="altcat_1.html#D10">ALTCAT_1:def 10</a></i>
;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:30"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="altcat_1.html#V11">associative</a>
 <b>from </b><i><a class="ref" href="yellow18.html#S2" target="_self">YELLOW18:sch 2</a>(<a class="txt" href="yellow18.html#E3:30"><i><font color="Green">E15</font></i></a>, <a class="txt" href="yellow18.html#E4:30"><i><font color="Green">E16</font></i></a>);</i><br/>


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