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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">A9</font>, <font color="Maroon" title="c4">B9</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_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"> for <font color="Olive" title="b1">F</font> being   <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>  st <font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="yellow20.html#R2" title="YELLOW20:pred.2">are_isomorphic_under</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be    <a href="functor0.html#L2" title="FUNCTOR0:struct.2">FunctorStr</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font>; <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="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="yellow20.html#R2" title="YELLOW20:pred.2">are_isomorphic_under</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/>
<font color="Maroon" title="c3">A9</font> is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E2:50"/>
<font color="Maroon" title="c4">B9</font> is   <a href="altcat_2.html#NM2" title="ALTCAT_2:NM.2">subcategory</a> of <font color="Maroon" title="c2">B</font>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow20.html#D4" target="_self" title="YELLOW20:def.4">YELLOW20:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( for <font color="Olive" title="b1">G</font> being  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font>  holds <br/>(  not <font color="Olive" title="b1">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  or  ex <font color="Olive" title="b2">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A9</font> ex <font color="Olive" title="b3">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a</font> &amp; not <font color="Olive" title="b1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b3">a</font> ) or  ex <font color="Olive" title="b2">b9</font>, <font color="Olive" title="b3">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A9</font> ex <font color="Olive" title="b4">b</font>, <font color="Olive" title="b5">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font> st <br/>( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b</font> &amp; <font color="Olive" title="b3">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">c</font> &amp;  ex <font color="Olive" title="b6">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b9</font>,<font color="Olive" title="b3">c9</font> ex <font color="Olive" title="b7">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font> st <br/>( <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">f</font> &amp; not <font color="Olive" title="b1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b6">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b4">b</font>,<font color="Olive" title="b5">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">f</font> ) ) ) or <font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a>  )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">G</font> being   <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>   <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon" title="c3">A9</font>,<font color="Maroon" title="c4">B9</font><span class="kw"> such that </span><a NAME="E4:50"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
<font color="Maroon" title="c6">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a> 
 <span class="kw">and </span><br/><a NAME="E5:50"/>
for <font color="Olive" title="b1">a9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A9</font><br/> for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> holds <br/><font color="Maroon" title="c6">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b1">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font>
 <span class="kw">and </span><br/><a NAME="E6:50"/>
for <font color="Olive" title="b1">b9</font>, <font color="Olive" title="b2">c9</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">A9</font><br/> for <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">A</font>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">b9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">c</font> holds <br/>for <font color="Olive" title="b5">f9</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">b9</font>,<font color="Olive" title="b2">c9</font><br/> for <font color="Olive" title="b6">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font>  st <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">f</font> holds <br/><font color="Maroon" title="c6">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b5">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="functor0.html#K4" title="FUNCTOR0:func.4">Morph-Map</a> <font color="Maroon" title="c5">F</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">f</font>
 ; <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="c3">A9</font>,<font color="Maroon" title="c4">B9</font> <a href="functor0.html#R1" title="FUNCTOR0:pred.1">are_isomorphic</a> </span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c6">G</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="functor0.html#D40" title="FUNCTOR0:def.40">FUNCTOR0:def 40</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="c6">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c6">G</font> is  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E7:50"/>
( <font color="Maroon" title="c6">G</font> is  <a href="functor0.html#V21" title="FUNCTOR0:attr.21">bijective</a>  &amp; <font color="Maroon" title="c6">G</font> is  <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50"><span class="lab"><font color="Green" title="E31">A1</font></span></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/>


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