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<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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#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>, <font color="Maroon" title="c5">F2</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">A</font>,<font color="Maroon" title="c3">B</font>;<br/><b>let </b><font color="Maroon" title="c6">G1</font>, <font color="Maroon" title="c7">G2</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="c3">B</font>,<font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c8">p</font> be    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>;<br/><b>let </b><font color="Maroon" title="c9">q</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">G1</font>,<font color="Maroon" title="c7">G2</font>;<br/>

















<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E21">A1</font></i>: 
( <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c5">F2</font> &amp; <font color="Maroon" title="c6">G1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">G2</font> )
 ;<br/>

<a NAME="E2:38"/><b>then </b><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c6">G1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G2</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E3:38"/><b>then </b><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c6">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">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>;<br/>
<a NAME="E4:38"/><i><font color="Green" title="E24">A4</font></i>: 
( <font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> &amp; <font color="Maroon" title="c6">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">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="functor3.html#E2:38"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="functor3.html#T10" target="_self" title="FUNCTOR3:th.10">Th10</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c6">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">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c4">F1</font> &amp; <font color="Maroon" title="c7">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">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font> )
 
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="functor3.html#E2:38"><i><font color="Green" title="E22">A2</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="38_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="c2">A</font>;<br/><a NAME="E1:38_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c6">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="c6">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">G1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</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">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</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">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="c7">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">G2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</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">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</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">sF2a</font> = <font color="Maroon" title="c9">q</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K1" title="FUNCTOR3:func.1">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span> ;<br/><b>reconsider </b><font color="Maroon" title="c12">G2ta</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</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="c7">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">G2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F2</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#E1:38_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><a NAME="E4:38_1"/><i><font color="Green" title="E27">A7</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><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 class="p2">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">a</font></span>)</span></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="functor3.html#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</font></span>)</span></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"><span class="p2">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c5">F2</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="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</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#E4:38"><i><font color="Green" title="E24">A4</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">sF2a</font> <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="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</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#E2:38"><i><font color="Green" title="E22">A2</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"><font color="Maroon" title="c9">q</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">F2</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="c6">G1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="functor3.html#E1:38_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self" title="FUNCTOR3:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</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="c9">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#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="functor3.html#E4:38_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">FUNCTOR2:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c12">G2ta</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#E1:38"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="functor3.html#T11" target="_self" title="FUNCTOR3:th.11">Th11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</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="c9">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#E2:38"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="functor3.html#E1:38_1"><i><font color="Green" title="E26">A6</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="c7">G2</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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#E5:38"><i><font color="Green" title="E25">A5</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>
<a NAME="E7:38"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c5">F2</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">G1</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">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#E3:38"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="functor2.html#T5" title="FUNCTOR2:th.5">FUNCTOR2:5</a></i>;<br/>
<b>hence </b><a NAME="E8:38"/>
<font color="Maroon" title="c9">q</font> <a href="functor3.html#K7" title="FUNCTOR3:func.7">(#)</a> <font color="Maroon" title="c8">p</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="functor3.html#K5" title="FUNCTOR3:func.5">*</a> <font color="Maroon" title="c8">p</font></span>)</span> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">q</font> <a href="functor3.html#K6" title="FUNCTOR3:func.6">*</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 ;<br/>


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