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<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a>  &amp; <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a>  implies <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a>  )</span><br/>





<span class="kw">given </span><font color="Maroon" title="c4">F1</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c5">G1</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c1">A</font><span class="kw"> such that </span><a NAME="E2:64"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
<font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E3:64"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">G1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">B</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="isocat_1.html#D10" target="_self" title="ISOCAT_1:def.10">ISOCAT_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a>  or <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">F2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">B</font>,<font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c7">G2</font> being    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">B</font><span class="kw"> such that </span><a NAME="E5:64"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
<font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E6:64"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c3">C</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="isocat_1.html#D10" target="_self" title="ISOCAT_1:def.10">ISOCAT_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c3">C</font> <a href="isocat_1.html#NR4" title="ISOCAT_1:NR.4">are_equivalent</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="isocat_1.html#D10" target="_self" title="ISOCAT_1:def.10">ISOCAT_1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">G</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">G</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c3">C</font> )</span><br/>

<a NAME="E7:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">G2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>;<br/>
<a NAME="E8:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A5</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="isocat_1.html#T49" target="_self" title="ISOCAT_1:th.49">Th49</a></span>;<br/>
<a NAME="E9:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c6">F2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>;<br/>
<a NAME="E10:64"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="isocat_1.html#T48" target="_self" title="ISOCAT_1:th.48">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="nattra_1.html#T32" title="NATTRA_1:th.32">NATTRA_1:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c3">C</font></span><br/>

<a NAME="E12:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">G1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">G1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>;<br/>
<a NAME="E13:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">G1</font> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">F2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="isocat_1.html#T49" target="_self" title="ISOCAT_1:th.49">Th49</a></span>;<br/>
<a NAME="E14:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c5">G1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></span>;<br/>
<a NAME="E15:64"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a> <font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:64"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="isocat_1.html#T48" target="_self" title="ISOCAT_1:th.48">Th48</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c4">F1</font></span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G1</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c7">G2</font></span>)</span> <a href="nattra_1.html#NR4" title="NATTRA_1:NR.4">~=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="nattra_1.html#T32" title="NATTRA_1:th.32">NATTRA_1:32</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
