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<div class="add">

<b>let </b><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">A</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</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">F1</font>, <font color="Olive" title="b2">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="c2">A</font>,<font color="Maroon" title="c1">B</font><br/> for <font color="Olive" title="b3">t</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font>  st <font color="Olive" title="b1">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b2">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Olive" title="b1">F1</font> &amp; ( for <font color="Olive" title="b4">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> holds  <font color="Olive" title="b3">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b4">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) holds <br/>( <font color="Olive" title="b2">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b1">F1</font> &amp;  ex <font color="Olive" title="b4">f</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b1">F1</font> st <br/>for <font color="Olive" title="b5">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> holds <br/> ( <font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b5">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Olive" title="b4">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b5">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c3">F1</font>, <font color="Maroon" title="c4">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="c1">B</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">t</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</font>  st <font color="Maroon" title="c3">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font> &amp; ( 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="c2">A</font> holds  <font color="Olive" title="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) holds <br/>( <font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font> &amp;  ex <font color="Olive" title="b2">f</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <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="c2">A</font> holds <br/> ( <font color="Olive" title="b2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="b1">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Olive" title="b2">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b3">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/><b>let </b><font color="Maroon" title="c5">t</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c3">F1</font>,<font color="Maroon" title="c4">F2</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">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F2</font> &amp; <font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font> &amp; ( 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="c2">A</font> holds  <font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) implies ( <font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font> &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <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="c2">A</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Olive" title="b1">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) ) )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c3">F1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green" title="E27">A3</font></i>: 
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="c2">A</font> holds  <font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font> &amp;  ex <font color="Olive" title="b1">f</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <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="c2">A</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Olive" title="b1">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> ) )</span><br/>

<b>set </b><font color="Maroon" title="c6">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <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="c2">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  );<br/>
<a NAME="E4:48"/><i><font color="Green" title="E28">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> holds <br/> ex <font color="Olive" title="b2">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> implies  ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>,<font color="Olive" title="b1">j</font>] )</span><br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">o</font> = <font color="Maroon" title="c7">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> ]</span><br/>

<b>thus </b><a NAME="E3:48_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">i</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c8">o</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> ]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c7">f</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green" title="E29">A5</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S3" title="PBOOLE:sch.3">PBOOLE:sch 3</a>(<a class="txt" href="functor3.html#E4:48"><i><font color="Green" title="E28">A4</font></i></a>);<br/></i>
<a NAME="E7:48"/><i><font color="Green" title="E30">A6</font></i>: 
<font color="Maroon" title="c3">F1</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c4">F2</font>
 
<b>by </b><i><a class="txt" href="functor3.html#E1:48"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a></i>;<br/>
<a NAME="E8:48"/>
<font color="Maroon" title="c7">f</font> is    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_2"/>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green" title="E26">A2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D2" title="FUNCTOR2:def.2">FUNCTOR2:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>) holds  <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> is   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> is   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>)</span><br/>

<b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E3:48_2"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="functor3.html#E6:48"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:48_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">a</font> is   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c8">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="functor3.html#E3:48_2"><i><font color="Green" title="E31">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">f</font> as    <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> ;<br/>
<div><i><font color="Green" title="E31">A8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></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>  implies for <font color="Olive" title="b1">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">g</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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span> )</span><br/><b>assume </b><a NAME="E1:48_3"/><i><font color="Green" title="E32">A9</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font></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> 
 ; <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">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b1">g</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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span></span><br/><a NAME="E2:48_3"/><i><font color="Green" title="E33">A10</font></i>: 
 ex <font color="Olive" title="b1">aa</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">a</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">aa</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="functor3.html#E6:48"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">fa</font> = <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">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="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span> ;<br/><a NAME="E4:48_3"/><i><font color="Green" title="E34">A11</font></i>: 
 ex <font color="Olive" title="b1">bb</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font> st <br/>( <font color="Olive" title="b1">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">bb</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="functor3.html#E6:48"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/><a NAME="E5:48_3"/><i><font color="Green" title="E35">A12</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><a NAME="E6:48_3"/><i><font color="Green" title="E36">A13</font></i>: 
<font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green" title="E27">A3</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c12">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c10">b</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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span></span><br/><a NAME="E7:48_3"/><i><font color="Green" title="E37">A14</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="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">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#E7:48"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E8:48_3"/><i><font color="Green" title="E38">A15</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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">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#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E9:48_3"/><i><font color="Green" title="E39">A16</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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</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#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E10:48_3"/><i><font color="Green" title="E40">A17</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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</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#E1:48_3"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></i>;<br/><a NAME="E11:48_3"/><b>then </b><i><font color="Green" title="E41">A18</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">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</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#E9:48_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></i>;<br/><a NAME="E12:48_3"/><i><font color="Green" title="E42">A19</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="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</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#E7:48"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/><a NAME="E13:48_3"/><i><font color="Green" title="E43">A20</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="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</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#E1:48_3"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="functor0.html#D19" title="FUNCTOR0:def.19">FUNCTOR0:def 19</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E11:48_3"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span></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="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c11">fa</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:48_3"><i><font color="Green" title="E33">A10</font></i></a>, <a class="txt" href="functor3.html#E5:48_3"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span></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="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E7:48_3"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="functor3.html#E8:48_3"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="functor3.html#E11:48_3"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span></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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E7:48_3"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="functor3.html#E9:48_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="functor3.html#E10:48_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</font></span>)</span></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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E1:48"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="functor3.html#E1:48_3"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">FUNCTOR2:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E9:48_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="functor3.html#E12:48_3"><i><font color="Green" title="E42">A19</font></i></a>, <a class="txt" href="functor3.html#E13:48_3"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="altcat_1.html#T25" title="ALTCAT_1:th.25">ALTCAT_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="functor3.html#E4:48_3"><i><font color="Green" title="E34">A11</font></i></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c10">b</font></span>)</span></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</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">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E6:48_3"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Maroon" title="c12">g</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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c9">a</font></span>)</span>
<b>by </b><i><a class="txt" href="functor3.html#E13:48_3"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="altcat_1.html#T24" title="ALTCAT_1:th.24">ALTCAT_1:24</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:48"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_4"/>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green" title="E26">A2</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D6" title="FUNCTOR2:def.6">FUNCTOR2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>3</sub></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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:48_4"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="functor2.html#M1" title="FUNCTOR2:mode.1">transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>4</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>3</sub></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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>4</sub></font></span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="functor3.html#E10:48"><i><font color="Green" title="E31">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E12:48"/>
<font color="Maroon" title="c8">f</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:48_5"/>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E11:48"><i><font color="Green" title="E32">A21</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="functor2.html#D7" title="FUNCTOR2:def.7">FUNCTOR2:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<b>thus </b><a NAME="E2:48_5"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</font>)  holds <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">M2</a>(<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>) holds  <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>2</sub></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="c4">F2</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F1</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive">b<sub>3</sub></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="c8">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="functor3.html#E10:48"><i><font color="Green" title="E31">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">f</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> ;<br/>
<b>thus </b><a NAME="E14:48"/>
<font color="Maroon" title="c4">F2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c3">F1</font>
 <b>by </b><i><a class="txt" href="functor3.html#E11:48"><i><font color="Green" title="E32">A21</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c4">F2</font>,<font color="Maroon" title="c3">F1</font> st <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="c2">A</font> holds <br/> ( <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Olive" title="b1">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b2">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><br/>

<b>take </b>
<font color="Maroon" title="c9">f</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="c2">A</font> holds <br/> ( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Olive" title="b1">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><br/>

<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  &amp; <font color="Maroon" title="c9">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><br/>

<b>consider </b><font color="Maroon" title="c11">b</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E16:48"/><i><font color="Green" title="E33">A22</font></i>: 
( <font color="Maroon" title="c11">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> &amp; <font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="functor3.html#E6:48"><i><font color="Green" title="E29">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E17:48"/>
<font color="Maroon" title="c9">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
 <b>by </b><i><a class="txt" href="functor3.html#E16:48"><i><font color="Green" title="E33">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<a NAME="E18:48"/><i><font color="Green" title="E34">A23</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="c3">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="c4">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>  &amp; <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">F2</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="c3">F1</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#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="functor3.html#E7:48"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="functor2.html#D1" title="FUNCTOR2:def.1">FUNCTOR2:def 1</a></i>;<br/>
<a NAME="E19:48"/>
<font color="Maroon" title="c9">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">t</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c11">b</font></span>)</span> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> 
 
<b>by </b><i><a class="txt" href="functor3.html#E2:48"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="functor3.html#E16:48"><i><font color="Green" title="E33">A22</font></i></a>, <a class="ref" href="functor2.html#D4" title="FUNCTOR2:def.4">FUNCTOR2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E20:48"/>
<font color="Maroon" title="c9">f</font> <a href="functor2.html#K2" title="FUNCTOR2:func.2">!</a> <font color="Maroon" title="c10">a</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="txt" href="functor3.html#E3:48"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="functor3.html#E16:48"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="functor3.html#E18:48"><i><font color="Green" title="E34">A23</font></i></a>, <a class="ref" href="altcat_4.html#T3" title="ALTCAT_4:th.3">ALTCAT_4:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
