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<b>let </b><font color="Maroon" title="c1">B</font>, <font color="Maroon" title="c2">C</font>, <font color="Maroon" title="c3">D</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b2">S</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">D</font>  st <font color="Olive" title="b1">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> &amp; <font color="Olive" title="b2">S</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> holds <br/><font color="Olive" title="b2">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Olive" title="b1">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a></span><br/><b>let </b><font color="Maroon" title="c4">T</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c1">B</font>,<font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being   <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">D</font>  st <font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> &amp; <font color="Olive" title="b1">S</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> holds <br/><font color="Olive" title="b1">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a></span><br/><b>let </b><font color="Maroon" title="c5">S</font> be    <a href="cat_1.html#M2" title="CAT_1:mode.2">Functor</a> of <font color="Maroon" title="c2">C</font>,<font color="Maroon" title="c3">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> &amp; <font color="Maroon" title="c5">S</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> implies <font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a> )</span><br/>









<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green" title="E60">A1</font></i>: 
<font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a>
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c5">S</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font> is <a href="cat_1.html#V11" title="CAT_1:attr.11">faithful</a></span><br/>

<b>let </b><font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">b'</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">B</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D24" target="_self" title="CAT_1:def.24">CAT_1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</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">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f2</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font> )</span><br/>

<b>assume </b><a NAME="E3:108"/><i><font color="Green" title="E62">A3</font></i>: 
 <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">f2</font> holds <br/><font color="Olive" title="b1">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f2</font></span><br/>

<b>let </b><font color="Maroon" title="c8">f1</font>, <font color="Maroon" title="c9">f2</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">b'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font> implies <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font> )</span><br/>

<b>assume </b><a NAME="E4:108"/><i><font color="Green" title="E63">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font></span><br/>

<a NAME="E5:108"/>
(  <a href="cat_1.html#K7" title="CAT_1:func.7">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font></span>)</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; <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">S</font> <a href="cat_1.html#K14" title="CAT_1:func.14">*</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font></span>)</span> &amp; <font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font> &amp; <font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font> is    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">T</font> <a href="cat_1.html#K13" title="CAT_1:func.13">.</a> <font color="Maroon" title="c7">b'</font> )
 
<b>by </b><i><a class="txt" href="cat_1.html#E3:108"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="cat_1.html#T125" target="_self" title="CAT_1:th.125">Th125</a>, <a class="ref" href="cat_1.html#T126" target="_self" title="CAT_1:th.126">Th126</a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E6:108"/><b>then </b>
<font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">f2</font>
 
<b>by </b><i><a class="txt" href="cat_1.html#E2:108"><i><font color="Green" title="E61">A2</font></i></a>, <a class="txt" href="cat_1.html#E4:108"><i><font color="Green" title="E63">A4</font></i></a>, <a class="ref" href="cat_1.html#D24" target="_self" title="CAT_1:def.24">Def24</a></i>;<br/>
<b>hence </b><a NAME="E7:108"/>
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f2</font>
 <b>by </b><i><a class="txt" href="cat_1.html#E1:108"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="cat_1.html#E3:108"><i><font color="Green" title="E62">A3</font></i></a>, <a class="ref" href="cat_1.html#D24" target="_self" title="CAT_1:def.24">Def24</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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