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<span class="kw">let </span><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">A</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"> for <font color="Olive" title="b1">F</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">B</font>  st <font color="Olive" title="b1">F</font> is  <a href="cat_1.html#V8" title="CAT_1:attr.8">isomorphic</a>  holds <br/>( <font color="Olive" title="b1">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">B</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">A</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c1">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">F</font> is  <a href="cat_1.html#V8" title="CAT_1:attr.8">isomorphic</a>  implies ( <font color="Maroon" title="c3">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">A</font> ) )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">f</font> = <font color="Maroon" title="c3">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">A</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">B</font> ;<br/>
<a NAME="E2:18"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:18"/><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="cat_1.html#V8" title="CAT_1:attr.8">isomorphic</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="c3">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">A</font> )</span><br/>

<a NAME="E4:18"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">Def3</a></span>;<br/>
<a NAME="E5:18"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="isocat_1.html#D3" target="_self" title="ISOCAT_1:def.3">Def3</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c3">F</font> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="isocat_1.html#D2" target="_self" title="ISOCAT_1:def.2">Def2</a></span>
<br/>.= 
 <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c1">B</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E5:18"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</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="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">A</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="isocat_1.html#K1" title="ISOCAT_1:func.1">"</a> </span>)</span> <a href="cat_1.html#K10" title="CAT_1:func.10">*</a> <font color="Maroon" title="c3">F</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:18"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="isocat_1.html#D2" target="_self" title="ISOCAT_1:def.2">Def2</a></span>
<br/>.= 
 <a href="cat_1.html#K11" title="CAT_1:func.11">id</a> <font color="Maroon" title="c2">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:18"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E2:18"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="funct_1.html#T61" title="FUNCT_1:th.61">FUNCT_1:61</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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