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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_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><br/>  for <font color="Olive" title="b3">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st <font color="Olive" title="b3">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a>  holds <br/>( <font color="Olive" title="b3">f</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  &amp; <font color="Olive" title="b3">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  )</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">  for <font color="Olive" title="b1">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a>  holds <br/>( <font color="Olive" title="b1">f</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">f</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="c4">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a>  implies ( <font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">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="c4">f</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  &amp; <font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  )</span><br/>

<a NAME="E2:49"/>
<font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V3" title="CAT_7:attr.3">section_</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:49"/>
<font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T28" target="_self" onmouseover="rs('cat_7/T28')" onmouseout="rh()">Th28</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"> <font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a> </span><br/>

<a NAME="E4:49"/>
<font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V4" title="CAT_7:attr.4">retraction</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E30">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:49"/>
<font color="Maroon" title="c4">f</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#T32" target="_self" onmouseover="rs('cat_7/T32')" onmouseout="rh()">Th32</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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