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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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#V12">with_units</a>  <a href="altcat_1.html#L2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be  <a href="functor0.html#V15">covariant</a>  <a href="functor0.html#M2">Functor</a> of <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="functor2.html#M1">transformation</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="functor2.html#M2">natural_transformation</a> of <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>;<br/>

















<b>assume </b><a NAME="E1:38"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R2">is_naturally_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="functor2.html#D6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E3:38"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor3.html#E2:38"><i><font color="Green">E22</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="functor2.html#R1">is_transformable_to</a> <font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor3.html#E2:38"><i><font color="Green">E22</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E1:38_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="functor0.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="functor0.html#T34">FUNCTOR0:34</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="functor2.html#K2">!</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K1" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="altcat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor0.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="functor3.html#E1:38_1"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E4:38_1"/><i><font color="Green">E27</font></i>: 
<span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></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="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor2.html#D1">FUNCTOR2:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E4:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="functor2.html#D5">FUNCTOR2:def 5</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:38"><i><font color="Green">E22</font></i></a>, <a class="ref" href="functor3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor2.html#K2">!</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor3.html#E1:38_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor0.html#K6">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor2.html#K2">!</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="txt" href="functor3.html#E4:38_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="functor2.html#D7">FUNCTOR2:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor2.html#K2">!</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="functor0.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green">E21</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="altcat_1.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:38"><i><font color="Green">E22</font></i></a>, <a class="txt" href="functor3.html#E1:38_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="functor3.html#T12" target="_self">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="functor2.html#K2">!</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="functor3.html#E5:38"><i><font color="Green">E25</font></i></a>, <a class="ref" href="functor2.html#D5">FUNCTOR2:def 5</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="functor3.html#E3:38"><i><font color="Green">E23</font></i></a>, <a class="ref" href="functor2.html#T5">FUNCTOR2:5</a></i>;<br/>
<b>hence </b><a NAME="E8:38"/>
<font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K7" target="_self">(#)</a> <font color="Maroon">c<sub>8</sub></font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="functor3.html#K5" target="_self">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="functor2.html#K3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="functor3.html#K6" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>


</div>
