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<span class="kw">let </span><font color="Maroon" title="c1">C</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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</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="b4">f1</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><br/>  for <font color="Olive" title="b5">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>  st <font color="Olive" title="b4">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  &amp; <font color="Olive" title="b5">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  holds <br/><font color="Olive" title="b5">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">f1</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>, <font color="Maroon" title="c4">c</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">f1</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><br/>  for <font color="Olive" title="b2">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>  st <font color="Olive" title="b1">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  &amp; <font color="Olive" title="b2">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  holds <br/><font color="Olive" title="b2">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">f1</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="c5">f1</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">  for <font color="Olive" title="b1">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>  st <font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  &amp; <font color="Olive" title="b1">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  holds <br/><font color="Olive" title="b1">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</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="c6">f2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">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"> ( <font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  &amp; <font color="Maroon" title="c6">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  implies <font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:43"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</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"> (  not <font color="Maroon" title="c6">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  or <font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:43"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c6">f2</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</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="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font> is  <a href="cat_7.html#V2" title="CAT_7:attr.2">epimorphism</a> </span><br/>

<span class="kw">hence </span><a NAME="E3:43"/>
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_7.html#D6" target="_self" onmouseover="rs('cat_7/D6')" onmouseout="rh()">CAT_7:def 6</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">c</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">c</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> for <font color="Olive" title="b2">g1</font>, <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">c</font>  st <font color="Olive" title="b2">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> holds <br/><font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">d</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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  for <font color="Olive" title="b1">g1</font>, <font color="Olive" title="b2">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:43"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">g1</font>, <font color="Olive" title="b2">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</font>  st <font color="Olive" title="b1">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> holds <br/><font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">g1</font>, <font color="Maroon" title="c9">g2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">d</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="c8">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> implies <font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:43"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c8">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</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="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font></span><br/>

<a NAME="E6:43"/><span class="lab"><font color="Green" title="E33">A5</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">d</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font> = 
<font color="Maroon" title="c8">g1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c6">f2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c5">f1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:43"/>
<font color="Maroon" title="c8">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E33">A5</font></span></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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