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<span class="kw">let </span><font color="Maroon" title="c1">C1</font>, <font color="Maroon" title="c2">C2</font> be   <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">F</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c1">C1</font>,<font color="Maroon" title="c2">C2</font>  st <font color="Olive" title="b1">F</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  holds <br/><font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</font> be   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <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"> ( <font color="Maroon" title="c3">F</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</a>  implies <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:9"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="cat_7.html#V9" title="CAT_7:attr.9">isomorphism</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"> <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><br/>

<a NAME="E2:9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D19" onmouseover="rs('cat_7/D19')" onmouseout="rh()">CAT_7:def 19</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">G</font> being   <a href="cat_6.html#NM4" title="CAT_6:NM.4">Functor</a> of <font color="Maroon" title="c2">C2</font>,<font color="Maroon" title="c1">C1</font><span class="kw"> such that </span><br/><a NAME="E4:9"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
( <font color="Maroon" title="c4">G</font> is  <a href="cat_6.html#V16" title="CAT_6:attr.16">covariant</a>  &amp; <font color="Maroon" title="c4">G</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c3">F</font> <a href="cat_6.html#K11" title="CAT_6:func.11">(*)</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="cat_7.html#D19" onmouseover="rs('cat_7/D19')" onmouseout="rh()">CAT_7:def 19</a></span>;<br/>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
( <font color="Maroon" title="c3">F</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c4">G</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font> &amp; <font color="Maroon" title="c4">G</font> <a href="relset_1.html#K4" title="RELSET_1:func.4">*</a> <font color="Maroon" title="c3">F</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="cat_6.html#D27" onmouseover="rs('cat_6/D27')" onmouseout="rh()">CAT_6:def 27</a></span>;<br/>
<a NAME="E6:9"/>
(  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c1">C1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C1</font> &amp;  <a href="cat_6.html#K12" title="CAT_6:func.12">id</a> <font color="Maroon" title="c2">C2</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">C2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="struct_0.html#D4" onmouseover="rs('struct_0/D4')" onmouseout="rh()">STRUCT_0:def 4</a></span>;<br/>
<a NAME="E7:9"/><span class="kw">then </span>
( <font color="Maroon" title="c3">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="funct_2.html#T23" onmouseover="rs('funct_2/T23')" onmouseout="rh()">FUNCT_2:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:9"/>
<font color="Maroon" title="c3">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</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"> verum</span><br/>


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