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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> being <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>  st <font color="Olive" title="b1">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F1</font> &amp; <font color="Olive" title="b2">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> holds <br/>for <font color="Olive" title="b4">t1</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">F1</font><br/> for <font color="Olive" title="b5">t2</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font><br/> for <font color="Olive" title="b6">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b5">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Olive" title="b4">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b6">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b6">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">F</font>, <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="c1">A</font>,<font color="Maroon" title="c2">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font> implies for <font color="Olive" title="b1">t1</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">F1</font><br/> for <font color="Olive" title="b2">t2</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Olive" title="b1">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F1</font>
 <b>and </b><br/><a NAME="E2:23"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c5">F2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t1</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">F1</font><br/> for <font color="Olive" title="b2">t2</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><br/> for <font color="Olive" title="b3">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Olive" title="b1">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c6">t1</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">F1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">t2</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font></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">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">t2</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F1</font>,<font color="Maroon" title="c5">F2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font></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">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c8">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font></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">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font></span>)</span></span><br/>





<a NAME="E3:23"/>
( <font color="Maroon" title="c3">F</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F1</font> &amp; <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> )
 
<b>by </b><i><a class="txt" href="#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E2:23"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="functor2.html#D6" target="_self" title="FUNCTOR2:def.6">Def6</a></i>;<br/>
<b>hence </b><a NAME="E4:23"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K3" title="FUNCTOR2:func.3">`*`</a> <font color="Maroon" title="c6">t1</font></span>)</span> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">t2</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font></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">t1</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="functor2.html#D5" target="_self" title="FUNCTOR2:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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