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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:107_1_1_1"/>
( (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A1</font></span>: 
(  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp; <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b4">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F1</font> : ( <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">car</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">car</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c5">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span> ;<br/><a NAME="E3:107_1_1_1_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="cat_8.html#E93"><span class="lab"><font color="Green" title="E59">Lm4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">comp</font> = <font color="Maroon" title="c6">comp</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="cat_8.html#E93"><span class="lab"><font color="Green" title="E59">Lm4</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">C</font> =  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c7">comp</font> #) as    <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</a>  ;<br/><a NAME="E7:107_1_1_1_1"/>
<font color="Maroon" title="c8">C</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_1"><span class="lab"><font color="Green" title="E61">A1</font></span></a>, <a class="txt" href="cat_8.html#E93"><span class="lab"><font color="Green" title="E59">Lm4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c8">C</font> as   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">C</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><span class="kw">thus </span><a NAME="E9:107_1_1_1_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c9">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">C</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A2</font></span>: 
( (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b4">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F1</font> : ( <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">car</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> ;<br/><a NAME="E2:107_1_1_1_2"/>
 not <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:107_1_1_1_2_1_1"/>
( (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2"><span class="lab"><font color="Green" title="E61">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_2_1_1_1"/>
(  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">CA</font> = <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c5">CB</font> = <font color="Maroon" title="c2">C2</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c6">F</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>;<br/><a NAME="E3:107_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A3</font></span>: 
 the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a>  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>, the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/><a NAME="E4:107_1_1_1_2_1_1_1"/>
 the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a>  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">T</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>, the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E62">A3</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/><a NAME="E6:107_1_1_1_2_1_1_1"/>
 the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a>  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/><a NAME="E7:107_1_1_1_2_1_1_1"/><span class="kw">then </span>
<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"> the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font>, the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c5">CB</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c7">T</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"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E8:107_1_1_1_2_1_1_1"/>
 not <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> </span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">CA</font> = <font color="Maroon" title="c1">C1</font> as   <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c5">F</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font>;<br/><a NAME="E3:107_1_1_1_2_1_1_2"/>
 the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a>  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">T</font> =  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font>, the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/><a NAME="E5:107_1_1_1_2_1_1_2"/>
 the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a>  the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/><a NAME="E6:107_1_1_1_2_1_1_2"/><span class="kw">then </span>
<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"> the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font>, the   <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c4">CA</font>,<font color="Maroon" title="c2">C2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c6">T</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"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span> 
 ;<br/><span class="kw">hence </span><a NAME="E7:107_1_1_1_2_1_1_2"/>
 not <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">car</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c5">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span> ;<br/><a NAME="E4:107_1_1_1_2"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  implies <font color="Maroon" title="c6">x</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">x2</font>, <font color="Maroon" title="c9">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_2"/><span class="lab"><font color="Green" title="E62">A4</font></span>: 
( <font color="Maroon" title="c6">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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c7">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 ;<br/>
<a NAME="E4:107_1_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x2</font>,<font color="Maroon" title="c7">x1</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:107_1_1_1_2_2"/>
<font color="Maroon" title="c6">x</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"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_2"><span class="lab"><font color="Green" title="E62">A4</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">comp</font> = <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  as   <a href="relset_1.html#NM1" title="RELSET_1:NM.1">Relation</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E6:107_1_1_1_2"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y1</font>, <font color="Maroon" title="c9">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font> implies <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font> or <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">x11</font>, <font color="Maroon" title="c11">x12</font>, <font color="Maroon" title="c12">x13</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E62">A5</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c11">x12</font>,<font color="Maroon" title="c10">x11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c12">x13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c10">x11</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x12</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">x13</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 ;<br/>
<span class="kw">assume </span><a NAME="E4:107_1_1_1_2_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">comp</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x21</font>, <font color="Maroon" title="c14">x22</font>, <font color="Maroon" title="c15">x23</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font><span class="kw"> such that </span><br/><a NAME="E6:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c9">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c14">x22</font>,<font color="Maroon" title="c13">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">x23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c13">x21</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">x22</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">x23</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 ;<br/>
<a NAME="E7:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E64">A7</font></span>: 
( <font color="Maroon" title="c7">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="c11">x12</font>,<font color="Maroon" title="c10">x11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x13</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E8:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E65">A8</font></span>: 
( <font color="Maroon" title="c7">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">x22</font>,<font color="Maroon" title="c13">x21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x23</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E66">A9</font></span>: 
( <font color="Maroon" title="c10">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x21</font> &amp; <font color="Maroon" title="c11">x12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E64">A7</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E65">A8</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">F11</font>, <font color="Maroon" title="c17">F12</font>, <font color="Maroon" title="c18">F13</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c19">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c16">F11</font>,<font color="Maroon" title="c17">F12</font>, <font color="Maroon" title="c20">T12</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c17">F12</font>,<font color="Maroon" title="c18">F13</font><span class="kw"> such that </span><br/><a NAME="E11:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E67">A10</font></span>: 
( <font color="Maroon" title="c10">x11</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F11</font>,<font color="Maroon" title="c17">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">x12</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F12</font>,<font color="Maroon" title="c18">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c20">T12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">x13</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F11</font>,<font color="Maroon" title="c18">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c20">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c19">T11</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E62">A5</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">F21</font>, <font color="Maroon" title="c22">F22</font>, <font color="Maroon" title="c23">F23</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c24">T21</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c21">F21</font>,<font color="Maroon" title="c22">F22</font>, <font color="Maroon" title="c25">T22</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c22">F22</font>,<font color="Maroon" title="c23">F23</font><span class="kw"> such that </span><br/><a NAME="E13:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E68">A11</font></span>: 
( <font color="Maroon" title="c13">x21</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">F21</font>,<font color="Maroon" title="c22">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c24">T21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">x22</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">F22</font>,<font color="Maroon" title="c23">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c25">T22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">x23</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">F21</font>,<font color="Maroon" title="c23">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c25">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c24">T21</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>;<br/>
<a NAME="E14:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E69">A12</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F11</font>,<font color="Maroon" title="c17">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c21">F21</font>,<font color="Maroon" title="c22">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c24">T21</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c19">T11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E13:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E15:107_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A13</font></span>: 
( <font color="Maroon" title="c16">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">F21</font> &amp; <font color="Maroon" title="c17">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c22">F22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E16:107_1_1_1_2_3"/><span class="lab"><font color="Green" title="E71">A14</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">F12</font>,<font color="Maroon" title="c18">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c22">F22</font>,<font color="Maroon" title="c23">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c25">T22</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">T12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E13:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E66">A9</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E17:107_1_1_1_2_3"/><span class="kw">then </span>
( <font color="Maroon" title="c17">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c22">F22</font> &amp; <font color="Maroon" title="c18">F13</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">F23</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:107_1_1_1_2_3"/>
<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a>, <a class="txt" href="#E6:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E63">A6</font></span></a>, <a class="txt" href="#E11:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E67">A10</font></span></a>, <a class="txt" href="#E13:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E68">A11</font></span></a>, <a class="txt" href="#E14:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E69">A12</font></span></a>, <a class="txt" href="#E15:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_3"><span class="lab"><font color="Green" title="E71">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">comp</font> = <font color="Maroon" title="c6">comp</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c4">car</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c4">car</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D1" onmouseover="rs('funct_1/D1')" onmouseout="rh()">FUNCT_1:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">C</font> =  <a href="cat_6.html#G1" title="CAT_6:aggr.1">CategoryStr</a>(# <font color="Maroon" title="c4">car</font>,<font color="Maroon" title="c7">comp</font> #) as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="cat_6.html#L1" title="CAT_6:struct.1">CategoryStr</a>  ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:107_1_1_1_2_4"/>
( (  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  ) or <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2"><span class="lab"><font color="Green" title="E61">A2</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_2_4_1"/>
(  not <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b4">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F1</font> : ( <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">CA</font> = <font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c10">CB</font> = <font color="Maroon" title="c2">C2</font> as   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ;<br/><a NAME="E3:107_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E62">A15</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> holds <br/> ex <font color="Olive" title="b3">F1</font>, <font color="Olive" title="b4">F2</font>, <font color="Olive" title="b5">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b6">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b3">F1</font>,<font color="Olive" title="b4">F2</font> ex <font color="Olive" title="b7">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F2</font>,<font color="Olive" title="b5">F3</font> st <br/>( <font color="Olive" title="b1">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F2</font>,<font color="Olive" title="b5">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F1</font>,<font color="Olive" title="b4">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b5">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b4">F2</font> &amp; <font color="Olive" title="b4">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b5">F3</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F1</font>,<font color="Olive" title="b5">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b7">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b6">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> implies  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b2">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E63">A16</font></span>: 
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b2">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E2:107_1_1_1_2_4_1_1"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c8">C</font>
 
;<br/>
<a NAME="E3:107_1_1_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">F11</font>, <font color="Maroon" title="c14">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c15">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c13">F11</font>,<font color="Maroon" title="c14">F12</font><span class="kw"> such that </span><br/><a NAME="E5:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E64">A17</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F11</font>,<font color="Maroon" title="c14">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F12</font> )
 ;<br/>
<a NAME="E6:107_1_1_1_2_4_1_1"/>
<font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_6.html#K1" title="CAT_6:func.1">Mor</a> <font color="Maroon" title="c8">C</font>
 
;<br/>
<a NAME="E7:107_1_1_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">F21</font>, <font color="Maroon" title="c17">F22</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c18">T22</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c16">F21</font>,<font color="Maroon" title="c17">F22</font><span class="kw"> such that </span><br/><a NAME="E9:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E65">A18</font></span>: 
( <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F21</font>,<font color="Maroon" title="c17">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c16">F21</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c17">F22</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F21</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c17">F22</font> )
 ;<br/>
<a NAME="E10:107_1_1_1_2_4_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c8">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E12:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E66">A19</font></span>: 
<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="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">comp</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">x1</font>, <font color="Maroon" title="c21">x2</font>, <font color="Maroon" title="c22">x3</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font><span class="kw"> such that </span><br/><a NAME="E14:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E67">A20</font></span>: 
( <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="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c19">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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="c21">x2</font>,<font color="Maroon" title="c20">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b4">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b5">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font> st <br/>( <font color="Maroon" title="c20">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b3">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b4">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E66">A19</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c23">F1</font>, <font color="Maroon" title="c24">F2</font>, <font color="Maroon" title="c25">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c26">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font>, <font color="Maroon" title="c27">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font><span class="kw"> such that </span><br/><a NAME="E16:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E68">A21</font></span>: 
( <font color="Maroon" title="c20">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c26">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c21">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c27">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E67">A20</font></span></a></span>;<br/>
<a NAME="E17:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E69">A22</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c21">x2</font>,<font color="Maroon" title="c20">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">x3</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E18:107_1_1_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A23</font></span>: 
( <font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x2</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c23">F1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F2</font>, <font color="Olive" title="b2">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b3">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c23">F1</font>,<font color="Olive" title="b1">F2</font> ex <font color="Olive" title="b4">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Olive" title="b1">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b1">F2</font> &amp; <font color="Olive" title="b1">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Olive" title="b2">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b4">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b3">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c24">F2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b2">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font> ex <font color="Olive" title="b3">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c24">F2</font>,<font color="Olive" title="b1">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Olive" title="b1">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b2">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b1">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Olive" title="b1">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b2">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c25">F3</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font> ex <font color="Olive" title="b2">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b2">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b2">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b1">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c26">T1</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b1">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c26">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c27">T2</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c27">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c26">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E19:107_1_1_1_2_4_1_1"/>
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c27">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c26">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="txt" href="#E17:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E69">A22</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<a NAME="E20:107_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E71">A24</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F11</font>,<font color="Maroon" title="c14">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c24">F2</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">F21</font>,<font color="Maroon" title="c17">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c24">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="txt" href="#E18:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E70">A23</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E21:107_1_1_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A25</font></span>: 
( <font color="Maroon" title="c13">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c14">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c16">F21</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">F1</font> &amp; <font color="Maroon" title="c17">F22</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">F2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E22:107_1_1_1_2_4_1_1"/>
( <font color="Maroon" title="c23">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F3</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E20:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E71">A24</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font> &amp; <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E23:107_1_1_1_2_4_1_1"/>
<font color="Maroon" title="c23">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c24">F2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E65">A18</font></span></a>, <a class="txt" href="#E21:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E72">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">thus </span><a NAME="E24:107_1_1_1_2_4_1_1"/>
<font color="Maroon" title="c24">F2</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A17</font></span></a>, <a class="txt" href="#E21:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E72">A25</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> = 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c8">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c11">g1</font>,<font color="Maroon" title="c12">g2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E63">A16</font></span></a>, <a class="ref" href="cat_6.html#D3" onmouseover="rs('cat_6/D3')" onmouseout="rh()">CAT_6:def 3</a></span>
<br/>.= 
 the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c8">C</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="c11">g1</font>,<font color="Maroon" title="c12">g2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c19">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E66">A19</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>
<br/>.= 
<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="c23">F1</font>,<font color="Maroon" title="c25">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c27">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c26">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E68">A21</font></span></a>, <a class="txt" href="#E14:107_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E67">A20</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:107_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E63">A26</font></span>: 
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st  ex <font color="Olive" title="b3">F1</font>, <font color="Olive" title="b4">F2</font>, <font color="Olive" title="b5">F3</font>, <font color="Olive" title="b6">F4</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b3">F1</font>,<font color="Olive" title="b4">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F3</font>,<font color="Olive" title="b6">F4</font> st <br/>( <font color="Olive" title="b1">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F1</font>,<font color="Olive" title="b4">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F3</font>,<font color="Olive" title="b6">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">F2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b5">F3</font> &amp; <font color="Olive" title="b3">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b6">F4</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b4">F2</font> &amp; <font color="Olive" title="b5">F3</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b6">F4</font> ) holds <br/><font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font>, <font color="Olive" title="b3">F3</font>, <font color="Olive" title="b4">F4</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font> ex <font color="Olive" title="b5">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font> ex <font color="Olive" title="b6">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b3">F3</font>,<font color="Olive" title="b4">F4</font> st <br/>( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b5">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">F3</font>,<font color="Olive" title="b4">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b6">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">F2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b3">F3</font> &amp; <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b4">F4</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> &amp; <font color="Olive" title="b3">F3</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b4">F4</font> ) implies <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c13">F1</font>, <font color="Maroon" title="c14">F2</font>, <font color="Maroon" title="c15">F3</font>, <font color="Maroon" title="c16">F4</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c17">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font>, <font color="Maroon" title="c18">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F3</font>,<font color="Maroon" title="c16">F4</font><span class="kw"> such that </span><a NAME="E2:107_1_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E64">A27</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c14">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F3</font>,<font color="Maroon" title="c16">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">F2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c15">F3</font> &amp; <font color="Maroon" title="c13">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F4</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c14">F2</font> &amp; <font color="Maroon" title="c15">F3</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F4</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font></span><br/>

<a NAME="E3:107_1_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
( <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">car</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">car</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">T2</font> = <font color="Maroon" title="c18">T2</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c14">F2</font>,<font color="Maroon" title="c16">F4</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">g3</font> = <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="c13">F1</font>,<font color="Maroon" title="c16">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E5:107_1_1_1_2_4_1_2"/>
<font color="Maroon" title="c13">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="ref" href="cat_8.html#T62" target="_self" onmouseover="rs('cat_8/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<a NAME="E6:107_1_1_1_2_4_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A29</font></span>: 
<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="c13">F1</font>,<font color="Maroon" title="c16">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_2"/>
<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="c12">g2</font>,<font color="Maroon" title="c11">g1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">F1</font>,<font color="Maroon" title="c16">F4</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c19">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E66">A29</font></span></a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">g2</font>,<font color="Maroon" title="c11">g1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:107_1_1_1_2_4_1_2"/>
<font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D2" onmouseover="rs('cat_6/D2')" onmouseout="rh()">CAT_6:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:107_1_1_1_2_4_1"/>
 for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b2">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g2</font> holds <br/>( <font color="Olive" title="b2">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> iff <font color="Olive" title="b3">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font> implies ( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> iff <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_3"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> iff <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">F11</font>, <font color="Maroon" title="c15">F12</font>, <font color="Maroon" title="c16">F13</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c17">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c15">F12</font>, <font color="Maroon" title="c18">T12</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F12</font>,<font color="Maroon" title="c16">F13</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_3"/><span class="lab"><font color="Green" title="E64">A30</font></span>: 
( <font color="Maroon" title="c12">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F12</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c15">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F13</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F12</font> &amp; <font color="Maroon" title="c15">F12</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F13</font> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T11</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> implies <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_3_1"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_3_1"/><span class="lab"><font color="Green" title="E65">A31</font></span>: 
( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/><a NAME="E4:107_1_1_1_2_4_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">T02</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_3_1"><span class="lab"><font color="Green" title="E65">A31</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><a NAME="E5:107_1_1_1_2_4_1_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c16">F13</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">F03</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:107_1_1_1_2_4_1_3_1"/>
<font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_3_1"><span class="lab"><font color="Green" title="E65">A31</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:107_1_1_1_2_4_1_3"/>
<font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E7:107_1_1_1_2_4_1_3"/><span class="lab"><font color="Green" title="E65">A32</font></span>: 
( <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c15">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">T11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">T02</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E7:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_4_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A33</font></span>: 
( <font color="Maroon" title="c14">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c15">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">F03</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E10:107_1_1_1_2_4_1_3"/>
<font color="Maroon" title="c14">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="ref" href="cat_8.html#T62" target="_self" onmouseover="rs('cat_8/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:107_1_1_1_2_4_1_3"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E64">A30</font></span></a>, <a class="txt" href="#E7:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E65">A32</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_3"><span class="lab"><font color="Green" title="E66">A33</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E6:107_1_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A34</font></span>: 
<font color="Maroon" title="c8">C</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/><a NAME="E7:107_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E65">A35</font></span>: 
 for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b2">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g2</font> holds <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g2</font> iff <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g1</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g</font>, <font color="Maroon" title="c12">g1</font>, <font color="Maroon" title="c13">g2</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font> implies ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> iff <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_4"/>
<font color="Maroon" title="c12">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> iff <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">F11</font>, <font color="Maroon" title="c15">F12</font>, <font color="Maroon" title="c16">F13</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c17">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c15">F12</font>, <font color="Maroon" title="c18">T12</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F12</font>,<font color="Maroon" title="c16">F13</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_4"/><span class="lab"><font color="Green" title="E66">A36</font></span>: 
( <font color="Maroon" title="c12">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F12</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c15">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F13</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F12</font> &amp; <font color="Maroon" title="c15">F12</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F13</font> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T11</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> implies <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_4_1"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_4_1"/><span class="lab"><font color="Green" title="E67">A37</font></span>: 
( <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <font color="Maroon" title="c11">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span>)</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="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/><a NAME="E4:107_1_1_1_2_4_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F11</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c22">T01</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_4_1"><span class="lab"><font color="Green" title="E67">A37</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><a NAME="E5:107_1_1_1_2_4_1_4_1"/><span class="kw">then </span>
( <font color="Maroon" title="c14">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c19">F01</font> &amp; <font color="Maroon" title="c16">F13</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">F02</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:107_1_1_1_2_4_1_4_1"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_4_1"><span class="lab"><font color="Green" title="E67">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:107_1_1_1_2_4_1_4"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E7:107_1_1_1_2_4_1_4"/><span class="lab"><font color="Green" title="E67">A38</font></span>: 
( <font color="Maroon" title="c11">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <font color="Maroon" title="c11">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_4"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F12</font>,<font color="Maroon" title="c16">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">T12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c22">T01</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E7:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E67">A38</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_4_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A39</font></span>: 
( <font color="Maroon" title="c15">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c19">F01</font> &amp; <font color="Maroon" title="c16">F13</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">F02</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E10:107_1_1_1_2_4_1_4"/>
<font color="Maroon" title="c14">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="ref" href="cat_8.html#T62" target="_self" onmouseover="rs('cat_8/T62')" onmouseout="rh()">Th62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:107_1_1_1_2_4_1_4"/>
<font color="Maroon" title="c11">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E66">A36</font></span></a>, <a class="txt" href="#E7:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E67">A38</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_4"><span class="lab"><font color="Green" title="E68">A39</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:107_1_1_1_2_4_1"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Olive" title="b2">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_5"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F11</font>, <font color="Maroon" title="c13">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c14">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c13">F12</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_5"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c12">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c13">F12</font> )
 ;<br/>
<a NAME="E4:107_1_1_1_2_4_1_5"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<font color="Maroon" title="c13">F12</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c13">F12</font>,<font color="Maroon" title="c13">F12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E5:107_1_1_1_2_4_1_5"/><span class="lab"><font color="Green" title="E68">A42</font></span>: 
<font color="Maroon" title="c13">F12</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c13">F12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c13">F12</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c13">F12</font>,<font color="Maroon" title="c13">F12</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">g</font> = <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="c13">F12</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_5"/>
<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="c13">F12</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E68">A42</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <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="c13">F12</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">g</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font> &amp; <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:107_1_1_1_2_4_1_5"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c11">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_5"><span class="lab"><font color="Green" title="E68">A42</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> </span><br/>

<a NAME="E10:107_1_1_1_2_4_1_5"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g1</font> holds <br/><font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">g1</font> implies <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_5_1"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c18">g1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_5_1"/><span class="lab"><font color="Green" title="E69">A43</font></span>: 
( <font color="Maroon" title="c17">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<a NAME="E4:107_1_1_1_2_4_1_5_1"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c13">F12</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">T</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">T02</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E5:107_1_1_1_2_4_1_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A45</font></span>: 
( <font color="Maroon" title="c20">F02</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">F12</font> &amp; <font color="Maroon" title="c21">F03</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c13">F12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E6:107_1_1_1_2_4_1_5_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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="c9">CA</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">T01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</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="c9">CA</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">T01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_5_1_1"/><span class="lab"><font color="Green" title="E72">A46</font></span>: 
<font color="Maroon" title="c24">x</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="c9">CA</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">T01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">f</font> = <font color="Maroon" title="c24">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E4:107_1_1_1_2_4_1_5_1_1"/><span class="lab"><font color="Green" title="E73">A47</font></span>: 
( <font color="Maroon" title="c26">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="cat_6.html#D6" onmouseover="rs('cat_6/D6')" onmouseout="rh()">CAT_6:def 6</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E6:107_1_1_1_2_4_1_5_1_1"/><span class="lab"><font color="Green" title="E74">A48</font></span>: 
( <font color="Maroon" title="c25">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E72">A46</font></span></a>, <a class="ref" href="cat_6.html#D7" onmouseover="rs('cat_6/D7')" onmouseout="rh()">CAT_6:def 7</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_5_1_1"/>
( <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  &amp; <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="ref" href="cat_6.html#T9" onmouseover="rs('cat_6/T9')" onmouseout="rh()">CAT_6:9</a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A49</font></span>: 
( <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_4_1_5_1_1"/><span class="lab"><font color="Green" title="E76">A50</font></span>: 
<font color="Maroon" title="c22">T01</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E10:107_1_1_1_2_4_1_5_1_1"/>
<font color="Maroon" title="c23">T02</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E11:107_1_1_1_2_4_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A51</font></span>: 
( <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c21">F03</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> &amp; <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F03</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E12:107_1_1_1_2_4_1_5_1_1"/>
<font color="Maroon" title="c23">T02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E13:107_1_1_1_2_4_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A52</font></span>: 
<font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="ref" href="cat_6.html#D22" onmouseover="rs('cat_6/D22')" onmouseout="rh()">CAT_6:def 22</a>, <a class="ref" href="cat_6.html#D25" onmouseover="rs('cat_6/D25')" onmouseout="rh()">CAT_6:def 25</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E78">A52</font></span></a>, <a class="txt" href="#E11:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E77">A51</font></span></a>, <a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>
<br/>.= 
<font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E76">A50</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E73">A47</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E74">A48</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_5_1_1"><span class="lab"><font color="Green" title="E75">A49</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>
<br/>.= 
<font color="Maroon" title="c22">T01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:107_1_1_1_2_4_1_5_1"/>
<font color="Maroon" title="c17">g</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c18">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_5_1"><span class="lab"><font color="Green" title="E71">A45</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:107_1_1_1_2_4_1_5"/>
<font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D4" onmouseover="rs('cat_6/D4')" onmouseout="rh()">CAT_6:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:107_1_1_1_2_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A53</font></span>: 
<font color="Maroon" title="c8">C</font> is  <a href="cat_6.html#V4" title="CAT_6:attr.4">with_left_identities</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D6" onmouseover="rs('cat_6/D6')" onmouseout="rh()">CAT_6:def 6</a></span>;<br/><a NAME="E10:107_1_1_1_2_4_1"/><span class="lab"><font color="Green" title="E67">A54</font></span>: 
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b1">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> holds <br/> ex <font color="Olive" title="b2">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g</font> &amp; <font color="Olive" title="b2">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font> implies  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_6"/>
<font color="Maroon" title="c11">g1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">C</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> st <br/>( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b1">g</font> &amp; <font color="Olive" title="b1">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">F11</font>, <font color="Maroon" title="c13">F12</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c14">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c13">F12</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_6"/><span class="lab"><font color="Green" title="E68">A55</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c13">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c14">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c13">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c12">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c13">F12</font> )
 ;<br/>
<a NAME="E4:107_1_1_1_2_4_1_6"/><span class="lab"><font color="Green" title="E69">A56</font></span>: 
<font color="Maroon" title="c12">F11</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c12">F11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<a NAME="E5:107_1_1_1_2_4_1_6"/><span class="lab"><font color="Green" title="E70">A57</font></span>: 
<font color="Maroon" title="c12">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c12">F11</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="ref" href="cat_8.html#T61" target="_self" onmouseover="rs('cat_8/T61')" onmouseout="rh()">Th61</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">T</font> = <font color="Maroon" title="c12">F11</font> as    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c12">F11</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E69">A56</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">g</font> = <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="c12">F11</font>,<font color="Maroon" title="c12">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_6"/>
<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="c12">F11</font>,<font color="Maroon" title="c12">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">car</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E70">A57</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">g</font> = <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="c12">F11</font>,<font color="Maroon" title="c12">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c15">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c17">g</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> &amp; <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E9:107_1_1_1_2_4_1_6"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E68">A55</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_6"><span class="lab"><font color="Green" title="E70">A57</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a> </span><br/>

<a NAME="E10:107_1_1_1_2_4_1_6"/>
 for <font color="Olive" title="b1">g1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">g1</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font> implies <font color="Maroon" title="c18">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_6_1"/>
<font color="Maroon" title="c18">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">g</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F01</font>, <font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c21">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c22">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>, <font color="Maroon" title="c23">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_6_1"/><span class="lab"><font color="Green" title="E71">A58</font></span>: 
( <font color="Maroon" title="c18">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F02</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F03</font> &amp; <font color="Maroon" title="c18">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c21">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<a NAME="E4:107_1_1_1_2_4_1_6_1"/><span class="lab"><font color="Green" title="E72">A59</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c12">F11</font>,<font color="Maroon" title="c12">F11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">T01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c15">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E5:107_1_1_1_2_4_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A60</font></span>: 
( <font color="Maroon" title="c19">F01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">F11</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c12">F11</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E6:107_1_1_1_2_4_1_6_1"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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="c9">CA</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">T02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_6_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c24">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">x</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="c9">CA</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">T02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_6_1_1"/><span class="lab"><font color="Green" title="E74">A61</font></span>: 
<font color="Maroon" title="c24">x</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="c9">CA</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">T02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c25">f</font> = <font color="Maroon" title="c24">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c26">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E4:107_1_1_1_2_4_1_6_1_1"/><span class="lab"><font color="Green" title="E75">A62</font></span>: 
( <font color="Maroon" title="c26">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="cat_6.html#D6" onmouseover="rs('cat_6/D6')" onmouseout="rh()">CAT_6:def 6</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c27">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E6:107_1_1_1_2_4_1_6_1_1"/><span class="lab"><font color="Green" title="E76">A63</font></span>: 
( <font color="Maroon" title="c25">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E74">A61</font></span></a>, <a class="ref" href="cat_6.html#D7" onmouseover="rs('cat_6/D7')" onmouseout="rh()">CAT_6:def 7</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_6_1_1"/>
( <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  &amp; <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="ref" href="cat_6.html#T9" onmouseover="rs('cat_6/T9')" onmouseout="rh()">CAT_6:9</a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A64</font></span>: 
( <font color="Maroon" title="c26">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c27">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_4_1_6_1_1"/>
<font color="Maroon" title="c22">T01</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c19">F01</font>,<font color="Maroon" title="c20">F02</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E10:107_1_1_1_2_4_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A65</font></span>: 
( <font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c19">F01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">F01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> &amp; <font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E11:107_1_1_1_2_4_1_6_1_1"/>
<font color="Maroon" title="c23">T02</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c20">F02</font>,<font color="Maroon" title="c21">F03</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E12:107_1_1_1_2_4_1_6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A66</font></span>: 
( <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> &amp; <font color="Maroon" title="c21">F03</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font> &amp; <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> &amp; <font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">F03</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T01</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c26">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c27">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E75">A62</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E76">A63</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E77">A64</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
<br/>.= 
<font color="Maroon" title="c23">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c25">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E72">A59</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E79">A66</font></span></a>, <a class="txt" href="#E10:107_1_1_1_2_4_1_6_1_1"><span class="lab"><font color="Green" title="E78">A65</font></span></a>, <a class="ref" href="cat_8.html#T1" target="_self" onmouseover="rs('cat_8/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<font color="Maroon" title="c23">T02</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:107_1_1_1_2_4_1_6_1"/>
<font color="Maroon" title="c18">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c17">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">g1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E71">A58</font></span></a>, <a class="txt" href="#E5:107_1_1_1_2_4_1_6_1"><span class="lab"><font color="Green" title="E73">A60</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:107_1_1_1_2_4_1_6"/>
<font color="Maroon" title="c17">g</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D5" onmouseover="rs('cat_6/D5')" onmouseout="rh()">CAT_6:def 5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E11:107_1_1_1_2_4_1"/>
 for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font>, <font color="Olive" title="b3">g3</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> &amp; <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g3</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b3">g3</font> &amp; <font color="Olive" title="b1">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Olive" title="b2">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font> holds <br/><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b2">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Olive" title="b3">g3</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">g1</font>, <font color="Maroon" title="c12">g2</font>, <font color="Maroon" title="c13">g3</font> be   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c8">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> implies <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_7"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">F01</font>, <font color="Maroon" title="c15">F02</font>, <font color="Maroon" title="c16">F03</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c17">T01</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c15">F02</font>, <font color="Maroon" title="c18">T02</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c15">F02</font>,<font color="Maroon" title="c16">F03</font><span class="kw"> such that </span><br/><a NAME="E3:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E68">A67</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F02</font>,<font color="Maroon" title="c16">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c18">T02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c15">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c17">T01</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c14">F01</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c15">F02</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c16">F03</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c14">F01</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c15">F02</font> &amp; <font color="Maroon" title="c15">F02</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c16">F03</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c16">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T01</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:107_1_1_1_2_4_1_7"/>
<font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font> or  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">F11</font>, <font color="Maroon" title="c20">F12</font>, <font color="Maroon" title="c21">F13</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c22">T11</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c20">F12</font>, <font color="Maroon" title="c23">T12</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c20">F12</font>,<font color="Maroon" title="c21">F13</font><span class="kw"> such that </span><br/><a NAME="E6:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E69">A68</font></span>: 
( <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">F12</font>,<font color="Maroon" title="c21">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c23">T12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c20">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c22">T11</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c19">F11</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c20">F12</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c21">F13</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c19">F11</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c20">F12</font> &amp; <font color="Maroon" title="c20">F12</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c21">F13</font> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c21">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T11</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E7:107_1_1_1_2_4_1_7"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c13">g3</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> or <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c24">F21</font>, <font color="Maroon" title="c25">F22</font>, <font color="Maroon" title="c26">F23</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c27">T21</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c24">F21</font>,<font color="Maroon" title="c25">F22</font>, <font color="Maroon" title="c28">T22</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c25">F22</font>,<font color="Maroon" title="c26">F23</font><span class="kw"> such that </span><br/><a NAME="E9:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E70">A69</font></span>: 
( <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c25">F22</font>,<font color="Maroon" title="c26">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c28">T22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c13">g3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F21</font>,<font color="Maroon" title="c25">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c27">T21</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c24">F21</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c25">F22</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c26">F23</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c24">F21</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c25">F22</font> &amp; <font color="Maroon" title="c25">F22</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c26">F23</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">F21</font>,<font color="Maroon" title="c26">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:107_1_1_1_2_4_1_7"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">F31</font>, <font color="Maroon" title="c30">F32</font>, <font color="Maroon" title="c31">F33</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c9">CA</font>,<font color="Maroon" title="c10">CB</font>, <font color="Maroon" title="c32">T31</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c29">F31</font>,<font color="Maroon" title="c30">F32</font>, <font color="Maroon" title="c33">T32</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Maroon" title="c30">F32</font>,<font color="Maroon" title="c31">F33</font><span class="kw"> such that </span><br/><a NAME="E12:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E71">A70</font></span>: 
( <font color="Maroon" title="c11">g1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c30">F32</font>,<font color="Maroon" title="c31">F33</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c33">T32</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c29">F31</font>,<font color="Maroon" title="c30">F32</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c32">T31</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c29">F31</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c30">F32</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c31">F33</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c29">F31</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c30">F32</font> &amp; <font color="Maroon" title="c30">F32</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Maroon" title="c31">F33</font> &amp; <font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</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="c29">F31</font>,<font color="Maroon" title="c31">F33</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E62">A15</font></span></a></span>;<br/>
<a NAME="E13:107_1_1_1_2_4_1_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">F02</font>,<font color="Maroon" title="c16">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c30">F32</font>,<font color="Maroon" title="c31">F33</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">T02</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c33">T32</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E14:107_1_1_1_2_4_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A71</font></span>: 
( <font color="Maroon" title="c15">F02</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c30">F32</font> &amp; <font color="Maroon" title="c16">F03</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c31">F33</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E15:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E73">A72</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c15">F02</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c20">F12</font>,<font color="Maroon" title="c21">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c17">T01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c23">T12</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E16:107_1_1_1_2_4_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A73</font></span>: 
( <font color="Maroon" title="c14">F01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c20">F12</font> &amp; <font color="Maroon" title="c15">F02</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c21">F13</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E17:107_1_1_1_2_4_1_7"/><span class="lab"><font color="Green" title="E75">A74</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c16">F03</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c25">F22</font>,<font color="Maroon" title="c26">F23</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c18">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c28">T22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E18:107_1_1_1_2_4_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A75</font></span>: 
( <font color="Maroon" title="c14">F01</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c25">F22</font> &amp; <font color="Maroon" title="c16">F03</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c26">F23</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E19:107_1_1_1_2_4_1_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c20">F12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c24">F21</font>,<font color="Maroon" title="c25">F22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c22">T11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c27">T21</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E20:107_1_1_1_2_4_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A76</font></span>: 
( <font color="Maroon" title="c19">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c24">F21</font> &amp; <font color="Maroon" title="c20">F12</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c25">F22</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E21:107_1_1_1_2_4_1_7"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c21">F13</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</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"><font color="Maroon" title="c29">F31</font>,<font color="Maroon" title="c30">F32</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c23">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c32">T31</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E22:107_1_1_1_2_4_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A77</font></span>: 
( <font color="Maroon" title="c19">F11</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c29">F31</font> &amp; <font color="Maroon" title="c21">F13</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c30">F32</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>;<br/>
<a NAME="E23:107_1_1_1_2_4_1_7"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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="c9">CA</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="107_1_1_1_2_4_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c34">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c34">x</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="c9">CA</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E79">A78</font></span>: 
<font color="Maroon" title="c34">x</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="c9">CA</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c35">f</font> = <font color="Maroon" title="c34">x</font> as   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E79">A78</font></span></a>, <a class="ref" href="cat_6.html#D1" onmouseover="rs('cat_6/D1')" onmouseout="rh()">CAT_6:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c36">f1</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E4:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E80">A79</font></span>: 
( <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c35">f</font> &amp; <font color="Maroon" title="c36">f1</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E79">A78</font></span></a>, <a class="ref" href="cat_6.html#D6" onmouseover="rs('cat_6/D6')" onmouseout="rh()">CAT_6:def 6</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c37">f2</font> being   <a href="cat_6.html#NM1" title="CAT_6:NM.1">morphism</a> of <font color="Maroon" title="c9">CA</font><span class="kw"> such that </span><br/><a NAME="E6:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E81">A80</font></span>: 
( <font color="Maroon" title="c35">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c37">f2</font> &amp; <font color="Maroon" title="c37">f2</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E79">A78</font></span></a>, <a class="ref" href="cat_6.html#D7" onmouseover="rs('cat_6/D7')" onmouseout="rh()">CAT_6:def 7</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a></span>;<br/>
<a NAME="E7:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E82">A81</font></span>: 
( <font color="Maroon" title="c37">f2</font> is  <a href="cat_6.html#V2" title="CAT_6:attr.2">left_identity</a>  &amp; <font color="Maroon" title="c36">f1</font> is  <a href="cat_6.html#V3" title="CAT_6:attr.3">right_identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="ref" href="cat_6.html#T9" onmouseover="rs('cat_6/T9')" onmouseout="rh()">CAT_6:9</a></span>;<br/>
<a NAME="E8:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A82</font></span>: 
( <font color="Maroon" title="c36">f1</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  &amp; <font color="Maroon" title="c37">f2</font> is  <a href="cat_6.html#V11" title="CAT_6:attr.11">identity</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E9:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E84">A83</font></span>: 
( <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c36">f1</font> &amp; <font color="Maroon" title="c37">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c37">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E82">A81</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a>, <a class="ref" href="cat_6.html#T24" onmouseover="rs('cat_6/T24')" onmouseout="rh()">CAT_6:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E85">A84</font></span>: <font color="Maroon" title="c32">T31</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c22">T11</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
;<br/>
<span class="lab"><font color="Green" title="E86">A85</font></span>: <font color="Maroon" title="c28">T22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c17">T01</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E84">A83</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
;<br/>
<a NAME="E12:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E87">A86</font></span>: 
<font color="Maroon" title="c18">T02</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c15">F02</font>,<font color="Maroon" title="c16">F03</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E13:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E88">A87</font></span>: 
<font color="Maroon" title="c17">T01</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c15">F02</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E14:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A88</font></span>: 
( <font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c14">F01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> &amp; <font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font> &amp; <font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">F01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> &amp; <font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E15:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E90">A89</font></span>: 
( <font color="Maroon" title="c10">CB</font> is  <a href="cat_6.html#V6" title="CAT_6:attr.6">left_composable</a>  &amp; <font color="Maroon" title="c10">CB</font> is  <a href="cat_6.html#V7" title="CAT_6:attr.7">right_composable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a></span>;<br/>
<a NAME="E16:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E91">A90</font></span>: 
( <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c36">f1</font> &amp; <font color="Maroon" title="c37">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c37">f2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E82">A81</font></span></a>, <a class="ref" href="cat_6.html#T24" onmouseover="rs('cat_6/T24')" onmouseout="rh()">CAT_6:24</a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="ref" href="cat_6.html#D14" onmouseover="rs('cat_6/D14')" onmouseout="rh()">CAT_6:def 14</a></span>;<br/>
<a NAME="E17:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E92">A91</font></span>: 
<font color="Maroon" title="c23">T12</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c20">F12</font>,<font color="Maroon" title="c21">F13</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E18:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A92</font></span>: 
<font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a></span>;<br/>
<a NAME="E19:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E94">A93</font></span>: 
<font color="Maroon" title="c17">T01</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c14">F01</font>,<font color="Maroon" title="c15">F02</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E20:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A94</font></span>: 
<font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a></span>;<br/>
<a NAME="E21:107_1_1_1_2_4_1_7_1"/>
<font color="Maroon" title="c22">T11</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c19">F11</font>,<font color="Maroon" title="c20">F12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E22:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A95</font></span>: 
<font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">T11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a></span>;<br/>
<a NAME="E23:107_1_1_1_2_4_1_7_1"/>
<font color="Maroon" title="c18">T02</font> <a href="cat_8.html#R3" title="CAT_8:pred.3">is_natural_transformation_of</a> <font color="Maroon" title="c15">F02</font>,<font color="Maroon" title="c16">F03</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="ref" href="cat_8.html#D26" target="_self" onmouseover="rs('cat_8/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<a NAME="E24:107_1_1_1_2_4_1_7_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A96</font></span>: 
<font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a></span>;<br/>
<a NAME="E25:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E98">A97</font></span>: 
<font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="txt" href="#E22:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="txt" href="#E15:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E73">A72</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a></span>;<br/>
<a NAME="E26:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E99">A98</font></span>: 
<font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E87">A86</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E74">A73</font></span></a>, <a class="txt" href="#E22:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a></span>;<br/>
<a NAME="E27:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E100">A99</font></span>: 
<font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E88">A87</font></span></a>, <a class="txt" href="#E18:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a></span>;<br/>
<a NAME="E28:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E101">A100</font></span>: 
<font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E98">A97</font></span></a>, <a class="txt" href="#E17:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E92">A91</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>;<br/>
<a NAME="E29:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E102">A101</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="txt" href="#E26:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E99">A98</font></span></a>, <a class="txt" href="#E28:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E101">A100</font></span></a>, <a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/>
<a NAME="E30:107_1_1_1_2_4_1_7_1"/><span class="lab"><font color="Green" title="E103">A102</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#NR2" title="CAT_6:NR.2">|&gt;</a> <font color="Maroon" title="c22">T11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E90">A89</font></span></a>, <a class="txt" href="#E18:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E93">A92</font></span></a>, <a class="txt" href="#E22:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E96">A95</font></span></a>, <a class="ref" href="cat_6.html#D8" onmouseover="rs('cat_6/D8')" onmouseout="rh()">CAT_6:def 8</a></span>;<br/>
<span class="lab"><font color="Green" title="E104">A103</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E92">A91</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E73">A72</font></span></a>, <a class="txt" href="#E22:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E89">A88</font></span></a>, <a class="txt" href="#E17:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E75">A74</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E19:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E94">A93</font></span></a>, <a class="txt" href="#E16:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E91">A90</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="ref" href="cat_8.html#T58" target="_self" onmouseover="rs('cat_8/T58')" onmouseout="rh()">Th58</a></span>
;<br/>
<span class="lab"><font color="Green" title="E105">A104</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E104">A103</font></span></a>, <a class="txt" href="#E26:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E99">A98</font></span></a>, <a class="txt" href="#E28:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E101">A100</font></span></a>, <a class="ref" href="cat_8.html#T1" target="_self" onmouseover="rs('cat_8/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E97">A96</font></span></a>, <a class="txt" href="#E20:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E95">A94</font></span></a>, <a class="txt" href="#E27:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E100">A99</font></span></a>, <a class="ref" href="cat_8.html#T2" target="_self" onmouseover="rs('cat_8/T2')" onmouseout="rh()">Th2</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c32">T31</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c33">T32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">T31</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">T11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E85">A84</font></span></a>, <a class="txt" href="#E3:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E68">A67</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c30">F32</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c23">T12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">F12</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">T11</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E102">A101</font></span></a>, <a class="txt" href="#E30:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E103">A102</font></span></a>, <a class="ref" href="cat_8.html#T1" target="_self" onmouseover="rs('cat_8/T1')" onmouseout="rh()">Th1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">T02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p4">(<span class="default"><font color="Maroon" title="c15">F02</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">T01</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c36">f1</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c25">F22</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font></span>)</span></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">T21</font> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c37">f2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E105">A104</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E69">A68</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span> <a href="cat_6.html#K9" title="CAT_6:func.9">.</a> <font color="Maroon" title="c35">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E80">A79</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E81">A80</font></span></a>, <a class="txt" href="#E8:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E83">A82</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="txt" href="#E11:107_1_1_1_2_4_1_7_1"><span class="lab"><font color="Green" title="E86">A85</font></span></a>, <a class="ref" href="cat_8.html#D27" target="_self" onmouseover="rs('cat_8/D27')" onmouseout="rh()">Def27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c28">T22</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Maroon" title="c27">T21</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_6.html#D21" onmouseover="rs('cat_6/D21')" onmouseout="rh()">CAT_6:def 21</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E24:107_1_1_1_2_4_1_7"/>
<font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">g2</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g1</font> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c12">g2</font></span>)</span> <a href="cat_6.html#K2" title="CAT_6:func.2">(*)</a> <font color="Maroon" title="c13">g3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E70">A69</font></span></a>, <a class="txt" href="#E12:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E71">A70</font></span></a>, <a class="txt" href="#E14:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E72">A71</font></span></a>, <a class="txt" href="#E18:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E76">A75</font></span></a>, <a class="txt" href="#E20:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E77">A76</font></span></a>, <a class="txt" href="#E22:107_1_1_1_2_4_1_7"><span class="lab"><font color="Green" title="E78">A77</font></span></a>, <a class="ref" href="funct_2.html#T12" onmouseover="rs('funct_2/T12')" onmouseout="rh()">FUNCT_2:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c8">C</font> as   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E67">A54</font></span></a>, <a class="txt" href="#E9:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E66">A53</font></span></a>, <a class="txt" href="#E7:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E65">A35</font></span></a>, <a class="txt" href="#E6:107_1_1_1_2_4_1"><span class="lab"><font color="Green" title="E64">A34</font></span></a>, <a class="ref" href="cat_6.html#D10" onmouseover="rs('cat_6/D10')" onmouseout="rh()">CAT_6:def 10</a>, <a class="ref" href="cat_6.html#D11" onmouseover="rs('cat_6/D11')" onmouseout="rh()">CAT_6:def 11</a>, <a class="ref" href="cat_6.html#D12" onmouseover="rs('cat_6/D12')" onmouseout="rh()">CAT_6:def 12</a>, <a class="ref" href="cat_6.html#D7" onmouseover="rs('cat_6/D7')" onmouseout="rh()">CAT_6:def 7</a>, <a class="ref" href="cat_6.html#D9" onmouseover="rs('cat_6/D9')" onmouseout="rh()">CAT_6:def 9</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c11">C</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><span class="kw">thus </span><a NAME="E13:107_1_1_1_2_4_1"/>
(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">F1</font>,<font color="Olive" title="b2">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">F1</font>, <font color="Olive" title="b2">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b3">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b1">F2</font>,<font color="Olive" title="b2">F1</font> : ( <font color="Olive" title="b1">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b2">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Maroon" title="c11">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">C</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="107_1_1_1_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:107_1_1_1_2_4_2"/><span class="lab"><font color="Green" title="E62">A105</font></span>: 
<font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b4">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F1</font> : ( <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <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"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E3:107_1_1_1_2_4_2"/>
( <font color="Maroon" title="c4">car</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b1">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b3">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font>, <font color="Olive" title="b3">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">car</font> :  ex <font color="Olive" title="b4">F1</font>, <font color="Olive" title="b5">F2</font>, <font color="Olive" title="b6">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b7">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font> ex <font color="Olive" title="b8">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font> st <br/>( <font color="Olive" title="b1">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b5">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b7">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F2</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">F1</font>,<font color="Olive" title="b6">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b8">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b7">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:107_1_1_1_2_4_2"><span class="lab"><font color="Green" title="E62">A105</font></span></a>, <a class="txt" href="cat_8.html#E94"><span class="lab"><font color="Green" title="E60">Lm5</font></span></a></span>;<br/><a NAME="E4:107_1_1_1_2_4_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">C</font> is   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T8" target="_self" onmouseover="rs('cat_8/T8')" onmouseout="rh()">Th8</a></span>;<br/><span class="kw">hence </span><a NAME="E5:107_1_1_1_2_4_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="cat_6.html#V1" title="CAT_6:attr.1">strict</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> st <br/>(  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">F1</font>,<font color="Olive" title="b3">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">T</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">F1</font>, <font color="Olive" title="b3">F2</font> is   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>, <font color="Olive" title="b4">T</font> is    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b2">F2</font>,<font color="Olive" title="b3">F1</font> : ( <font color="Olive" title="b2">F1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b3">F2</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F1</font> <a href="cat_8.html#R4" title="CAT_8:pred.4">is_naturally_transformable_to</a> <font color="Olive" title="b3">F2</font> ) </span> } </span>  &amp;  the <a href="cat_6.html#U1" title="CAT_6:sel.1">composition</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x2</font>,<font color="Olive" title="b2">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b4">x3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b2">x1</font>, <font color="Olive" title="b3">x2</font>, <font color="Olive" title="b4">x3</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> :  ex <font color="Olive" title="b5">F1</font>, <font color="Olive" title="b6">F2</font>, <font color="Olive" title="b7">F3</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> ex <font color="Olive" title="b8">T1</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font> ex <font color="Olive" title="b9">T2</font> being    <a href="cat_8.html#M3" title="CAT_8:mode.3">natural_transformation</a> of <font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font> st <br/>( <font color="Olive" title="b2">x1</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b6">F2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b8">T1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">x2</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b6">F2</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<font color="Olive" title="b9">T2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b4">x3</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"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b5">F1</font>,<font color="Olive" title="b7">F3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Olive" title="b9">T2</font> <a href="cat_8.html#K13" title="CAT_8:func.13">`*`</a> <font color="Olive" title="b8">T1</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) </span> } </span>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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