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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_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="E1:131_1_1_1"/>
( <font color="Maroon" title="c1">C1</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  or ( <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <font color="Maroon" title="c1">C1</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>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:131_1_1_1_1"/><span class="lab"><font color="Green" title="E76">A1</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="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ex <font color="Olive" title="b2">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b3">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <font color="Olive">b<sub>1</sub></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="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">E</font> =  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Maroon" title="c2">C2</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1_1_1"><span class="lab"><font color="Green" title="E76">A1</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">IT</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><span class="kw">take </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.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">  ex <font color="Olive" title="b1">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b2">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</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="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
 <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1
; <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">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Olive" title="b1">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Olive" title="b1">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c4">E</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">thus </span><a NAME="E3:131_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> 1,<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1_1_1"><span class="lab"><font color="Green" title="E76">A1</font></span></a>, <a class="txt" href="cat_8.html#E113"><span class="lab"><font color="Green" title="E75">Lm8</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:131_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A2</font></span>: 
( <font color="Maroon" title="c2">C2</font> is  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  &amp;  not <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="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ex <font color="Olive" title="b2">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b3">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <font color="Olive">b<sub>1</sub></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="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c4">E</font> =  <a href="cat_8.html#K3" title="CAT_8:func.3">OrdC0-&gt;</a> <font color="Maroon" title="c2">C2</font> as   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c5">IT</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><span class="kw">take </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.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">  ex <font color="Olive" title="b1">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b2">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</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="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
 <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</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" title="b1">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Olive" title="b1">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Olive" title="b1">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c4">E</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">thus </span><a NAME="E3:131_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_7.html#K6" title="CAT_7:func.6">OrdC</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1_1_2"><span class="lab"><font color="Green" title="E76">A2</font></span></a>, <a class="txt" href="cat_8.html#E112"><span class="lab"><font color="Green" title="E74">Lm7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="131_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:131_1_1_1_3"/><span class="lab"><font color="Green" title="E76">A3</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>  )
 ; <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="xtuple_0.html#V1" title="XTUPLE_0:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ex <font color="Olive" title="b2">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b3">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <font color="Olive">b<sub>1</sub></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="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b3">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">C</font>,<font color="Olive" title="b3">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">C</font> =  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>);<br/><span class="kw">consider </span><font color="Maroon" title="c4">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font><span class="kw"> such that </span><br/><a NAME="E3:131_1_1_1_3"/><span class="lab"><font color="Green" title="E77">A4</font></span>: 
<font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 <span class="kw">and </span><br/><a NAME="E4:131_1_1_1_3"/><span class="lab"><font color="Green" title="E78">A5</font></span>: 
 for <font color="Olive" title="b1">D</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a><br/>  for <font color="Olive" title="b2">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">D</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b2">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  holds <br/> ex <font color="Olive" title="b3">H</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> st <br/>( <font color="Olive" title="b3">H</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">H</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">H1</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Olive" title="b1">D</font>,<span class="p1">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>  st <font color="Olive" title="b4">H1</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b2">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c4">E</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">H1</font> <a href="cat_8.html#K11" title="CAT_8:func.11">[x]</a> <span class="p2">(<span class="default"><a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b3">H</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Olive" title="b4">H1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:131_1_1_1_3"><span class="lab"><font color="Green" title="E76">A3</font></span></a>, <a class="txt" href="cat_8.html#E111"><span class="lab"><font color="Green" title="E73">Lm6</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">IT</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/><span class="kw">take </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.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">  ex <font color="Olive" title="b1">C</font> being   <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a> ex <font color="Olive" title="b2">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">C</font> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</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="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b2">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
 <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</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">E</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span> <a href="cat_8.html#K8" title="CAT_8:func.8">[x]</a> <font color="Maroon" title="c1">C1</font></span>)</span>,<font color="Maroon" title="c2">C2</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Olive" title="b1">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Olive" title="b1">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>),<font color="Olive" title="b1">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c4">E</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>),<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )</span><br/><span class="kw">thus </span><a NAME="E5:131_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</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">(<span class="default"><a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>)</span>)</span>,<font color="Maroon" title="c4">E</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c4">E</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp;  <a href="cat_8.html#K14" title="CAT_8:func.14">Functors</a> (<font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>),<font color="Maroon" title="c4">E</font> <a href="cat_8.html#R6" title="CAT_8:pred.6">is_exponent_of</a> <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131_1_1_1_3"><span class="lab"><font color="Green" title="E78">A5</font></span></a>, <a class="txt" href="#E3:131_1_1_1_3"><span class="lab"><font color="Green" title="E77">A4</font></span></a>, <a class="ref" href="cat_8.html#D34" target="_self" onmouseover="rs('cat_8/D34')" onmouseout="rh()">Def34</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></div><span class="kw">end;</span></div>

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