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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a> <a href="altcat_1.html#V12" title="ALTCAT_1:attr.12">with_units</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a> ;<br/><b>let </b><font color="Maroon" title="c4">F1</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/><b>let </b><font color="Maroon" title="c5">G1</font>, <font color="Maroon" title="c6">G2</font>, <font color="Maroon" title="c7">G3</font> be  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>;<br/><b>let </b><font color="Maroon" title="c8">q</font> be    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font>;<br/><b>let </b><font color="Maroon" title="c9">q1</font> be    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c7">G3</font>;<br/>

















<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c5">G1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G2</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G3</font> )
 ;<br/>

<a NAME="E2:29"/><b>then </b>
<font color="Maroon" title="c5">G1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G3</font>
 
<b>by </b><i><a class="ref" href="functor2.html#T4" title="FUNCTOR2:th.4">FUNCTOR2:4</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G3</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font>
 
<b>by </b><i><a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>;<br/>
<a NAME="E4:29"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G3</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>;<br/><a NAME="E1:29_1"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c5">G1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c7">G3</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G3</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="ref" href="functor0.html#T34" title="FUNCTOR0:th.34">FUNCTOR0:34</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">s1F1a</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">G2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">G3</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> ;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="functor3.html#T12" target="_self" title="FUNCTOR3:th.12">Th12</a>, <a class="ref" href="functor2.html#T4" title="FUNCTOR2:th.4">FUNCTOR2:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="functor2.html#D5" title="FUNCTOR2:def.5">FUNCTOR2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c11">s1F1a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="functor3.html#T12" target="_self" title="FUNCTOR3:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:29"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="functor3.html#E1:29_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="functor3.html#T12" target="_self" title="FUNCTOR3:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font>
<b>by </b><i><a class="txt" href="functor3.html#E4:29"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="functor2.html#D5" title="FUNCTOR2:def.5">FUNCTOR2:def 5</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:29"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q1</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <b>by </b><i><a class="txt" href="functor3.html#E3:29"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="functor2.html#T5" title="FUNCTOR2:th.5">FUNCTOR2:5</a></i>;<br/>


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