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<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>for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $2 iff ( <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">g</font> &amp; <font color="Olive" title="b3">x</font> is    <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">g</font> ) );<br/>
<a NAME="E1:27_1_1"/><i><font color="Green" title="E21">A1</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> 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="27_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> 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="c3">i</font>,<font color="Olive" title="b1">j</font>] )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ; <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="c3">i</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<b>then consider </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c5">g</font> as  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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> <b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</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">o</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> st <br/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">o</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">f</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">o</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> );<br/>
<a NAME="E5:27_1_1_1"/><i><font color="Green" title="E23">A3</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  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>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">j</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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">  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>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">j</font>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c9">o</font> = <font color="Maroon" title="c8">a</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<b>consider </b><font color="Maroon" title="c10">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">f</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">o</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c9">o</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 ;<br/>
<b>thus </b><a NAME="E4:27_1_1_1_1"/>
 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>2</sub></font>[<font color="Maroon" title="c8">a</font>,<font color="Olive" title="b1">j</font>]
 <b>by </b><i><a class="txt" href="#E3:27_1_1_1_1"><i><font color="Green" title="E24">A4</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>consider </b><font color="Maroon" title="c8">N</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="c1">A</font><b> such that </b><br/><a NAME="E7:27_1_1_1"/><i><font color="Green" title="E24">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>]
 <b>from </b><i><a class="ref" href="pboole.html#S6" title="PBOOLE:sch.6">PBOOLE:sch 6</a>(<a class="txt" href="#E5:27_1_1_1"><i><font color="Green" title="E23">A3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c6">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c7">g</font> &amp; $1 is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font> );<br/>
<b>consider </b><font color="Maroon" title="c9">j</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E9:27_1_1_1"/><i><font color="Green" title="E25">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">N</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c9">j</font>
; <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="c3">i</font>,<font color="Maroon" title="c9">j</font>]</span><br/>

<b>let </b><font color="Maroon" title="c10">f'</font>, <font color="Maroon" title="c11">g'</font> be  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> iff ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> ) ) )</span><br/>

<b>assume </b><a NAME="E10:27_1_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> iff ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c12">x</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="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> iff ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> ) )</span><br/>

<b>thus </b><a NAME="E11:27_1_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> implies ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_1_2"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> )</span><br/>

<a NAME="E2:27_1_1_1_2"/>
( <font color="Maroon" title="c10">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c11">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E10:27_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_1_2"/>
( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> )
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E1:27_1_1_1_2"><i><font color="Green" title="E27">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>thus </b><a NAME="E12:27_1_1_1"/>
( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> implies <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_1_3"/><i><font color="Green" title="E27">A9</font></i>: 
( <font color="Maroon" title="c10">f'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">g'</font> &amp; <font color="Maroon" title="c12">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">f'</font>,<font color="Maroon" title="c11">g'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font></span><br/>

<a NAME="E2:27_1_1_1_3"/><b>then </b><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c10">f'</font> <a href="functor2.html#R1" title="FUNCTOR2:pred.1">is_transformable_to</a> <font color="Maroon" title="c11">g'</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D6" target="_self" title="FUNCTOR2:def.6">Def6</a></i>;<br/>
<a NAME="E3:27_1_1_1_3"/><i><font color="Green" title="E29">A11</font></i>: 
( <font color="Maroon" title="c10">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c11">g'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">g</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E10:27_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">h</font> = <font color="Maroon" title="c12">x</font> as    <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="c1">A</font> <b>by </b><i><a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/>
<a NAME="E5:27_1_1_1_3"/><i><font color="Green" title="E30">A12</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">I</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>;<br/>
<a NAME="E6:27_1_1_1_3"/><i><font color="Green" title="E31">A13</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="c1">A</font> holds <br/><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="c15">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="c1">A</font> implies <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_3_1"/>
<font color="Maroon" title="c15">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="c1">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="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">i'</font> = <font color="Maroon" title="c15">i'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> ;<br/>
<b>consider </b><font color="Maroon" title="c17">i''</font> being   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E4:27_1_1_1_3_1"/><i><font color="Green" title="E32">A14</font></i>: 
<font color="Maroon" title="c17">i''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i'</font>
 <b>and </b><br/><a NAME="E5:27_1_1_1_3_1"/><i><font color="Green" title="E33">A15</font></i>: 
<font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">f</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E7:27_1_1_1"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E6:27_1_1_1_3_1"/><i><font color="Green" title="E34">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="c6">f</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</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="#E2:27_1_1_1_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="functor2.html#D1" target="_self" title="FUNCTOR2:def.1">Def1</a></i>;<br/>
<a NAME="E7:27_1_1_1_3_1"/>
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i''</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="c6">f</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Maroon" title="c17">i''</font></span>)</span></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<b>by </b><i><a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="#E2:27_1_1_1_3"><i><font color="Green" title="E28">A10</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="functor2.html#D2" target="_self" title="FUNCTOR2:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1_1_3_1"/>
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_1_3_1"><i><font color="Green" title="E32">A14</font></i></a>, <a class="txt" href="#E5:27_1_1_1_3_1"><i><font color="Green" title="E33">A15</font></i></a>, <a class="txt" href="#E6:27_1_1_1_3_1"><i><font color="Green" title="E34">A16</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="E7:27_1_1_1_3"/><i><font color="Green" title="E32">A17</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>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">N</font> holds <br/><font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">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="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">N</font> implies <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_1_3_2"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font></span><br/>

<a NAME="E2:27_1_1_1_3_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_1_3_2"/>
<font color="Maroon" title="c13">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">N</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i'</font>
 <b>by </b><i><a class="txt" href="#E6:27_1_1_1_3"><i><font color="Green" title="E31">A13</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3_2"><i><font color="Green" title="E33">A18</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="E8:27_1_1_1_3"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="txt" href="#E5:27_1_1_1_3"><i><font color="Green" title="E30">A12</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:27_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <font color="Maroon" title="c8">N</font>
 
<b>by </b><i><a class="txt" href="#E7:27_1_1_1_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<b>hence </b><a NAME="E10:27_1_1_1_3"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">j</font>
 <b>by </b><i><a class="txt" href="#E9:27_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="#E1:27_1_1_1_3"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="#E3:27_1_1_1_3"><i><font color="Green" title="E29">A11</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>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">a</font> being    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E22">A19</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> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c3">a</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="#E1:27_1_1"><i><font color="Green" title="E21">A1</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <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>for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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 <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $3 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">g</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">g</font>,<font color="Olive" title="b3">h</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">t2</font>,<font color="Olive" title="b4">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp;  ex <font color="Olive" title="b6">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b6">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b4">t1</font>;<br/>
<a NAME="E4:27_1_1"/><i><font color="Green" title="E23">A20</font></i>: 
for <font color="Olive" title="b1">o</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> holds <br/> ex <font color="Olive" title="b3">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">y</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">o</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">o</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="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">o</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_2"/>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 ; <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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">o</font>] )</span><br/>

<a NAME="E2:27_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p3">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">ff</font>, <font color="Maroon" title="c6">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:27_1_1_2"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c5">ff</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E5:27_1_1_2"/><i><font color="Green" title="E25">A22</font></i>: 
<font color="Maroon" title="c6">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E6:27_1_1_2"/><i><font color="Green" title="E26">A23</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">ff</font>,<font color="Maroon" title="c6">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:27_1_1_2"/><i><font color="Green" title="E27">A24</font></i>: 
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E9:27_1_1_2"/><i><font color="Green" title="E28">A25</font></i>: 
<font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E10:27_1_1_2"/><i><font color="Green" title="E29">A26</font></i>: 
<font color="Maroon" title="c5">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_2"><i><font color="Green" title="E24">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E11:27_1_1_2"/><i><font color="Green" title="E30">A27</font></i>: 
<font color="Maroon" title="c4">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c6">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="#E6:27_1_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="#E10:27_1_1_2"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">T</font> =  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E5:27_1_1_2"><i><font color="Green" title="E25">A22</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">i</font> = <font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c11">j</font> = <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c12">k</font> = <font color="Maroon" title="c6">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">T</font> <b>by </b><i><a class="txt" href="#E5:27_1_1_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="#E8:27_1_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="#E9:27_1_1_2"><i><font color="Green" title="E28">A25</font></i></a></i>;<br/>
<i><font color="Green" title="E31">A28</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>
;<br/>
<i><font color="Green" title="E32">A29</font></i>: <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c12">k</font>
<b>by </b><i><a class="ref" href="altcat_1.html#D5" title="ALTCAT_1:def.5">ALTCAT_1:def 5</a></i>
;<br/>
<a NAME="E16:27_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">o</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> implies  ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c4">o</font>] ) )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_2_3"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</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">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c4">o</font>] )</span><br/>

<b>then consider </b><font color="Maroon" title="c14">x2</font>, <font color="Maroon" title="c15">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1_2_3"/><i><font color="Green" title="E33">A30</font></i>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font>
 <b>and </b><br/><a NAME="E4:27_1_1_2_3"/><i><font color="Green" title="E34">A31</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font>
 <b>and </b><br/><a NAME="E5:27_1_1_2_3"/><i><font color="Green" title="E35">A32</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x2</font>,<font color="Maroon" title="c15">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E11:27_1_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="#E14:27_1_1_2"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E6:27_1_1_2_3"/><i><font color="Green" title="E36">A33</font></i>: 
<font color="Maroon" title="c14">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c11">j</font>,<font color="Maroon" title="c12">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1_2_3"><i><font color="Green" title="E33">A30</font></i></a></i>;<br/>
<a NAME="E7:27_1_1_2_3"/><i><font color="Green" title="E37">A34</font></i>: 
<font color="Maroon" title="c15">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c11">j</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="#E4:27_1_1_2_3"><i><font color="Green" title="E34">A31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">i'</font> = <font color="Maroon" title="c10">i</font>, <font color="Maroon" title="c17">j'</font> = <font color="Maroon" title="c11">j</font>, <font color="Maroon" title="c18">k'</font> = <font color="Maroon" title="c12">k</font> as  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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> <b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<a NAME="E9:27_1_1_2_3"/><i><font color="Green" title="E38">A35</font></i>: 
( <font color="Maroon" title="c16">i'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c17">j'</font> &amp; <font color="Maroon" title="c15">x1</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c17">j'</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E7:27_1_1_2_3"><i><font color="Green" title="E37">A34</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">x1</font> = <font color="Maroon" title="c15">x1</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c17">j'</font> <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E7:27_1_1_2_3"><i><font color="Green" title="E37">A34</font></i></a></i>;<br/>
<a NAME="E11:27_1_1_2_3"/><i><font color="Green" title="E39">A36</font></i>: 
( <font color="Maroon" title="c17">j'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c18">k'</font> &amp; <font color="Maroon" title="c14">x2</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">j'</font>,<font color="Maroon" title="c18">k'</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E6:27_1_1_2_3"><i><font color="Green" title="E36">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">x2</font> = <font color="Maroon" title="c14">x2</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c17">j'</font>,<font color="Maroon" title="c18">k'</font> <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E6:27_1_1_2_3"><i><font color="Green" title="E36">A33</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">vv</font> = <font color="Maroon" title="c20">x2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c19">x1</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c18">k'</font> ;<br/>
<a NAME="E14:27_1_1_2_3"/><i><font color="Green" title="E40">A37</font></i>: 
( <font color="Maroon" title="c16">i'</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c18">k'</font> &amp; <font color="Maroon" title="c21">vv</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c18">k'</font> )
 
<b>by </b><i><a class="txt" href="#E9:27_1_1_2_3"><i><font color="Green" title="E38">A35</font></i></a>, <a class="txt" href="#E11:27_1_1_2_3"><i><font color="Green" title="E39">A36</font></i></a>, <a class="ref" href="functor2.html#T10" target="_self" title="FUNCTOR2:th.10">Th10</a></i>;<br/>
<a NAME="E15:27_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c18">k'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E16:27_1_1_2_3"/><b>then </b><i><font color="Green" title="E41">A38</font></i>: 
<font color="Maroon" title="c21">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i'</font>,<font color="Maroon" title="c18">k'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E14:27_1_1_2_3"><i><font color="Green" title="E40">A37</font></i></a></i>;<br/>
<a NAME="E17:27_1_1_2_3"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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 <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</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">g</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">g</font>,<font color="Olive" title="b3">h</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">t2</font>,<font color="Olive" title="b4">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  ex <font color="Olive" title="b6">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b6">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font> holds <br/><font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b5">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b4">t1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">f</font>, <font color="Maroon" title="c23">g</font>, <font color="Maroon" title="c24">h</font> be  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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"> ( <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c22">f</font>,<font color="Maroon" title="c23">g</font>,<font color="Maroon" title="c24">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</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="c22">f</font>,<font color="Maroon" title="c23">g</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="c23">g</font>,<font color="Maroon" title="c24">h</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">t2</font>,<font color="Olive" title="b1">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  ex <font color="Olive" title="b3">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font> holds <br/><font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t1</font> )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_2_3_1"/><i><font color="Green" title="E42">A39</font></i>: 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c22">f</font>,<font color="Maroon" title="c23">g</font>,<font color="Maroon" title="c24">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">o</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="c22">f</font>,<font color="Maroon" title="c23">g</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="c23">g</font>,<font color="Maroon" title="c24">h</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">t2</font>,<font color="Olive" title="b1">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  ex <font color="Olive" title="b3">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b3">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font> holds <br/><font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b2">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t1</font></span><br/>

<b>let </b><font color="Maroon" title="c25">t1</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c22">f</font>,<font color="Maroon" title="c23">g</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="c23">g</font>,<font color="Maroon" title="c24">h</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">t2</font>,<font color="Maroon" title="c25">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  ex <font color="Olive" title="b2">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font> holds <br/><font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c25">t1</font></span><br/><b>let </b><font color="Maroon" title="c26">t2</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c23">g</font>,<font color="Maroon" title="c24">h</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="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">t2</font>,<font color="Maroon" title="c25">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> &amp;  ex <font color="Olive" title="b1">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font> implies <font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c25">t1</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E2:27_1_1_2_3_1"/><i><font color="Green" title="E43">A40</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">t2</font>,<font color="Maroon" title="c25">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font>
 <b>and </b><br/><a NAME="E3:27_1_1_2_3_1"/>
 ex <font color="Olive" title="b1">v</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <font color="Olive" title="b1">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">vv</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c25">t1</font></span><br/>

<a NAME="E4:27_1_1_2_3_1"/><i><font color="Green" title="E44">A41</font></i>: 
( <font color="Maroon" title="c16">i'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f</font> &amp; <font color="Maroon" title="c17">j'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">g</font> &amp; <font color="Maroon" title="c18">k'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">h</font> )
 
<b>by </b><i><a class="txt" href="#E11:27_1_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="#E1:27_1_1_2_3_1"><i><font color="Green" title="E42">A39</font></i></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c27">x1</font> = <font color="Maroon" title="c19">x1</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c22">f</font>,<font color="Maroon" title="c23">g</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c28">x2</font> = <font color="Maroon" title="c20">x2</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c23">g</font>,<font color="Maroon" title="c24">h</font> <b>by </b><i><a class="txt" href="#E4:27_1_1_2_3_1"><i><font color="Green" title="E44">A41</font></i></a></i>;<br/>
<a NAME="E7:27_1_1_2_3_1"/>
( <font color="Maroon" title="c28">x2</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">t2</font> &amp; <font color="Maroon" title="c27">x1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c25">t1</font> )
 
<b>by </b><i><a class="txt" href="#E5:27_1_1_2_3"><i><font color="Green" title="E35">A32</font></i></a>, <a class="txt" href="#E2:27_1_1_2_3_1"><i><font color="Green" title="E43">A40</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1_2_3_1"/>
<font color="Maroon" title="c21">vv</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c26">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c25">t1</font>
 <b>by </b><i><a class="txt" href="#E4:27_1_1_2_3_1"><i><font color="Green" title="E44">A41</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>hence </b><a NAME="E18:27_1_1_2_3"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c13">x</font>,<font color="Maroon" title="c4">o</font>] )
 <b>by </b><i><a class="txt" href="#E11:27_1_1_2"><i><font color="Green" title="E30">A27</font></i></a>, <a class="txt" href="#E15:27_1_1_2"><i><font color="Green" title="E32">A29</font></i></a>, <a class="txt" href="#E16:27_1_1_2_3"><i><font color="Green" title="E41">A38</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>hence </b><a NAME="E17:27_1_1_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">o</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">o</font>] )
 ; <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="c4">c</font> being    <a href="pboole.html#M3" title="PBOOLE:mode.3">ManySortedFunction</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span><b> such that </b><br/><a NAME="E6:27_1_1"/><i><font color="Green" title="E24">A42</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> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds <br/> ex <font color="Olive" title="b2">v</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> st <br/>( <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1" title="MSSUBFAM:sch.1">MSSUBFAM:sch 1</a>(<a class="txt" href="#E4:27_1_1"><i><font color="Green" title="E23">A20</font></i></a>);<br/></i>
<b>set </b><font color="Maroon" title="c5">F</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> #);<br/>
<a NAME="E7:27_1_1"/><i><font color="Green" title="E25">A43</font></i>: 
not  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_3"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">f</font> being  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>;<br/>
<a NAME="E2:27_1_1_3"/>
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1_1_3"/>
not  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</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>
<a NAME="E8:27_1_1"/>
 <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> #) is <a href="altcat_1.html#V2" title="ALTCAT_1:attr.2">transitive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c7">o2</font>, <font color="Maroon" title="c8">o3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a> <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="c6">o1</font>,<font color="Maroon" title="c7">o2</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 <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c8">o3</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  not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c8">o3</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>  )</span><br/>

<b>assume </b><a NAME="E1:27_1_1_4"/><i><font color="Green" title="E26">A44</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="c6">o1</font>,<font color="Maroon" title="c7">o2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c8">o3</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"> not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c8">o3</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> </span><br/>

<b>reconsider </b><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c6">o1</font>, <font color="Maroon" title="c10">a2</font> = <font color="Maroon" title="c7">o2</font>, <font color="Maroon" title="c11">a3</font> = <font color="Maroon" title="c8">o3</font> as  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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> <b>by </b><i><a class="txt" href="#E7:27_1_1"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<a NAME="E3:27_1_1_4"/><i><font color="Green" title="E27">A45</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c7">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E7:27_1_1"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:27_1_1_4"/><i><font color="Green" title="E28">A46</font></i>: 
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c7">o2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E6:27_1_1_4"/><i><font color="Green" title="E29">A47</font></i>: 
( <font color="Maroon" title="c9">a1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c10">a2</font> &amp; <font color="Maroon" title="c12">y</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">a2</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E3:27_1_1_4"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="#E5:27_1_1_4"><i><font color="Green" title="E28">A46</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">y</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">a2</font> <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E3:27_1_1_4"><i><font color="Green" title="E27">A45</font></i></a>, <a class="txt" href="#E5:27_1_1_4"><i><font color="Green" title="E28">A46</font></i></a></i>;<br/>
<a NAME="E8:27_1_1_4"/><i><font color="Green" title="E30">A48</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c8">o3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E7:27_1_1"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:27_1_1_4"/><i><font color="Green" title="E31">A49</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">o2</font>,<font color="Maroon" title="c8">o3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="#E1:27_1_1_4"><i><font color="Green" title="E26">A44</font></i></a>, <a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E11:27_1_1_4"/><i><font color="Green" title="E32">A50</font></i>: 
( <font color="Maroon" title="c10">a2</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">a3</font> &amp; <font color="Maroon" title="c14">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">a3</font> )
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E8:27_1_1_4"><i><font color="Green" title="E30">A48</font></i></a>, <a class="txt" href="#E10:27_1_1_4"><i><font color="Green" title="E31">A49</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c10">a2</font>,<font color="Maroon" title="c11">a3</font> <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E8:27_1_1_4"><i><font color="Green" title="E30">A48</font></i></a>, <a class="txt" href="#E10:27_1_1_4"><i><font color="Green" title="E31">A49</font></i></a></i>;<br/>
<a NAME="E13:27_1_1_4"/><i><font color="Green" title="E33">A51</font></i>: 
<font color="Maroon" title="c15">x</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c13">y</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c11">a3</font>
 
;<br/>
<a NAME="E14:27_1_1_4"/><i><font color="Green" title="E34">A52</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c8">o3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E7:27_1_1"><i><font color="Green" title="E25">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<a NAME="E15:27_1_1_4"/>
( <font color="Maroon" title="c9">a1</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c11">a3</font> &amp;  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Olive" title="b1">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c11">a3</font> )
 
<b>by </b><i><a class="txt" href="#E6:27_1_1_4"><i><font color="Green" title="E29">A47</font></i></a>, <a class="txt" href="#E11:27_1_1_4"><i><font color="Green" title="E32">A50</font></i></a>, <a class="txt" href="#E13:27_1_1_4"><i><font color="Green" title="E33">A51</font></i></a>, <a class="ref" href="functor2.html#T10" target="_self" title="FUNCTOR2:th.10">Th10</a></i>;<br/>
<b>hence </b><a NAME="E16:27_1_1_4"/>
not <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">o1</font>,<font color="Maroon" title="c8">o3</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> 
 <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E14:27_1_1_4"><i><font color="Green" title="E34">A52</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="c6">F</font> =  <a href="altcat_1.html#G2" title="ALTCAT_1:aggr.2">AltCatStr</a>(# <span class="p1">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font> #) as  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#V6" title="ALTCAT_1:attr.6">strict</a>  <a href="altcat_1.html#L2" title="ALTCAT_1:struct.2">AltCatStr</a>  <b>by </b><i><a class="txt" href="#E7:27_1_1"><i><font color="Green" title="E25">A43</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; ( for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font> iff ( <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">G</font> &amp; <font color="Olive" title="b3">x</font> is    <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">G</font> ) ) ) &amp; ( for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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">G</font> &amp; <font color="Olive" title="b2">G</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">H</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">G</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">G</font>,<font color="Olive" title="b3">H</font>  ex <font color="Olive" title="b6">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b6">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">t2</font>,<font color="Olive" title="b4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b4">t1</font> ) ) )</span><br/>

<b>thus </b><a NAME="E10:27_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <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"> ( ( for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font> iff ( <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">G</font> &amp; <font color="Olive" title="b3">x</font> is    <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">G</font> ) ) ) &amp; ( for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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">G</font> &amp; <font color="Olive" title="b2">G</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">H</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">G</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">G</font>,<font color="Olive" title="b3">H</font>  ex <font color="Olive" title="b6">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b6">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">t2</font>,<font color="Olive" title="b4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b4">t1</font> ) ) )</span><br/>

<b>thus </b><a NAME="E11:27_1_1"/>
for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font><br/> for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font> iff ( <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">g</font> &amp; <font color="Olive" title="b3">x</font> is    <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">g</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">F</font>, <font color="Olive" title="b2">G</font>, <font color="Olive" title="b3">H</font> being <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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">G</font> &amp; <font color="Olive" title="b2">G</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Olive" title="b3">H</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">G</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">G</font>,<font color="Olive" title="b3">H</font>  ex <font color="Olive" title="b6">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Olive" title="b1">F</font>,<font color="Olive" title="b2">G</font>,<font color="Olive" title="b3">H</font> &amp; <font color="Olive" title="b6">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b5">t2</font>,<font color="Olive" title="b4">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b4">t1</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font> be  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> iff ( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Olive" title="b1">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> ) )</span><br/><b>let </b><font color="Maroon" title="c9">x</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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> iff ( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> ) )</span><br/>



<b>thus </b><a NAME="E1:27_1_1_5"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> implies ( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</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="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_5_1"/><i><font color="Green" title="E26">A53</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</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="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> )</span><br/>

<a NAME="E2:27_1_1_5_1"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<a NAME="E3:27_1_1_5_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1_1_5_1"/>
( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> )
 <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E1:27_1_1_5_1"><i><font color="Green" title="E26">A53</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>thus </b><a NAME="E2:27_1_1_5"/>
( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1_5_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1_5_2"/><i><font color="Green" title="E26">A54</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c9">x</font> is    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><br/>

<a NAME="E2:27_1_1_5_2"/>
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<a NAME="E3:27_1_1_5_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1_1_5_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <font color="Maroon" title="c6">F</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>
 <b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E1:27_1_1_5_2"><i><font color="Green" title="E26">A54</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>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c9">h</font> be  <a href="functor0.html#V9" title="FUNCTOR0:attr.9">strict</a> <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="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">h</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="c7">f</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c9">h</font>  ex <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">t2</font>,<font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t1</font> ) )</span><br/>

<b>assume </b><a NAME="E12:27_1_1"/><i><font color="Green" title="E26">A55</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c8">g</font> <a href="functor2.html#R2" title="FUNCTOR2:pred.2">is_naturally_transformable_to</a> <font color="Maroon" title="c9">h</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="c7">f</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c9">h</font>  ex <font color="Olive" title="b3">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b3">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b2">t2</font>,<font color="Olive" title="b1">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Olive" title="b1">t1</font> )</span><br/>

<b>let </b><font color="Maroon" title="c10">t1</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</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="c8">g</font>,<font color="Maroon" title="c9">h</font>  ex <font color="Olive" title="b2">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b2">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">t2</font>,<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c10">t1</font> )</span><br/><b>let </b><font color="Maroon" title="c11">t2</font> be    <a href="functor2.html#M2" title="FUNCTOR2:mode.2">natural_transformation</a> of <font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</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">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">t2</font>,<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c10">t1</font> )</span><br/>



<a NAME="E13:27_1_1"/><i><font color="Green" title="E27">A56</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c8">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c9">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<a NAME="E14:27_1_1"/><i><font color="Green" title="E28">A57</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p3">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3" title="ZFMISC_1:def.3">ZFMISC_1:def 3</a></i>;<br/>
<a NAME="E15:27_1_1"/><i><font color="Green" title="E29">A58</font></i>: 
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D3" title="MCART_1:def.3">MCART_1:def 3</a></i>;<br/>
<a NAME="E16:27_1_1"/><i><font color="Green" title="E30">A59</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E13:27_1_1"><i><font color="Green" title="E27">A56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:27_1_1"/><i><font color="Green" title="E31">A60</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E13:27_1_1"><i><font color="Green" title="E27">A56</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E18:27_1_1"/><b>then </b>
<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span>,<span class="p2">(<span class="default"><a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>
 
<b>by </b><i><a class="txt" href="#E13:27_1_1"><i><font color="Green" title="E27">A56</font></i></a>, <a class="txt" href="#E14:27_1_1"><i><font color="Green" title="E28">A57</font></i></a>, <a class="txt" href="#E15:27_1_1"><i><font color="Green" title="E29">A58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">v</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>,<span class="p1"><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">{|</a><span class="default"><font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K2" title="ALTCAT_1:func.2">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span><b> such that </b><br/><a NAME="E20:27_1_1"/><i><font color="Green" title="E32">A61</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <b>and </b><br/><a NAME="E21:27_1_1"/><i><font color="Green" title="E33">A62</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b1">x</font>,<span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>]
 <b>by </b><i><a class="txt" href="#E6:27_1_1"><i><font color="Green" title="E24">A42</font></i></a></i>;<br/>
<a NAME="E22:27_1_1"/><i><font color="Green" title="E34">A63</font></i>: 
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font>
 
<b>by </b><i><a class="txt" href="#E20:27_1_1"><i><font color="Green" title="E32">A61</font></i></a>, <a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">T</font> =  <a href="functor2.html#K7" title="FUNCTOR2:func.7">Funct</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="#E13:27_1_1"><i><font color="Green" title="E27">A56</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c7">f</font>, <font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c8">g</font>, <font color="Maroon" title="c16">k</font> = <font color="Maroon" title="c9">h</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c13">T</font> <b>by </b><i><a class="ref" href="functor2.html#D10" target="_self" title="FUNCTOR2:def.10">Def10</a></i>;<br/>
<i><font color="Green" title="E35">A64</font></i>: <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c16">k</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> = 
<span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c16">k</font>
<b>by </b><i><a class="ref" href="multop_1.html#D1" title="MULTOP_1:def.1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">j</font>,<font color="Maroon" title="c16">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c14">i</font>,<font color="Maroon" title="c15">j</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
<b>by </b><i><a class="ref" href="altcat_1.html#D6" title="ALTCAT_1:def.6">ALTCAT_1:def 6</a></i>
;<br/>
<a NAME="E26:27_1_1"/><i><font color="Green" title="E36">A65</font></i>: 
<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E12:27_1_1"><i><font color="Green" title="E26">A55</font></i></a>, <a class="txt" href="#E17:27_1_1"><i><font color="Green" title="E31">A60</font></i></a></i>;<br/>
<a NAME="E27:27_1_1"/>
<font color="Maroon" title="c11">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="#E3:27_1_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="#E12:27_1_1"><i><font color="Green" title="E26">A55</font></i></a>, <a class="txt" href="#E16:27_1_1"><i><font color="Green" title="E30">A59</font></i></a></i>;<br/>
<a NAME="E28:27_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">t2</font>,<font color="Maroon" title="c10">t1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">{|</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K3" title="ALTCAT_1:func.3">|}</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 
<b>by </b><i><a class="txt" href="#E25:27_1_1"><i><font color="Green" title="E35">A64</font></i></a>, <a class="txt" href="#E26:27_1_1"><i><font color="Green" title="E36">A65</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E29:27_1_1"/><b>then </b>
<font color="Maroon" title="c12">v</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">t2</font>,<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c10">t1</font>
 
<b>by </b><i><a class="txt" href="#E21:27_1_1"><i><font color="Green" title="E33">A62</font></i></a></i>;<br/>
<b>hence </b><a NAME="E30:27_1_1"/>
 ex <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c6">F</font> <a href="multop_1.html#K1" title="MULTOP_1:func.1">.</a> <font color="Maroon" title="c7">f</font>,<font color="Maroon" title="c8">g</font>,<font color="Maroon" title="c9">h</font> &amp; <font color="Olive" title="b1">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">t2</font>,<font color="Maroon" title="c10">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t2</font> <a href="functor2.html#K5" title="FUNCTOR2:func.5">`*`</a> <font color="Maroon" title="c10">t1</font> )
 <b>by </b><i><a class="txt" href="#E22:27_1_1"><i><font color="Green" title="E34">A63</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>
