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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="cat_8.html#V7" title="CAT_8:attr.7">with_binary_products</a>   <a href="cat_8.html#V8" title="CAT_8:attr.8">with_exponential_objects</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</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">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> holds <br/> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Olive" title="b1">a</font>, <a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">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"> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>, <a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c4">T</font> =  the    <a href="cat_8.html#M4" title="CAT_8:mode.4">categorical_exponent</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">c</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c6">e</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><span class="kw"> such that </span><br/><a NAME="E2:125"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
(  the    <a href="cat_8.html#M4" title="CAT_8:mode.4">categorical_exponent</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#D31" target="_self" onmouseover="rs('cat_8/D31')" onmouseout="rh()">Def31</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:125"/>
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>, <a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:125"><span class="lab"><font color="Green" title="E72">A1</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/>


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