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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">c</font>, <font color="Olive" title="b2">c1</font>, <font color="Olive" title="b3">c2</font>, <font color="Olive" title="b4">d</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="b5">f1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b2">c1</font>,<font color="Olive" title="b1">c</font><br/>  for <font color="Olive" title="b6">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">c2</font>,<font color="Olive" title="b1">c</font><br/>  for <font color="Olive" title="b7">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">d</font>,<font color="Olive" title="b2">c1</font><br/>  for <font color="Olive" title="b8">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b2">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b3">c2</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b4">d</font>,<font color="Olive" title="b2">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c2</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>  &amp; <font color="Olive" title="b4">d</font>,<font color="Olive" title="b7">p1</font>,<font color="Olive" title="b8">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font> &amp; <font color="Olive" title="b5">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  holds <br/><font color="Olive" title="b8">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">c1</font>, <font color="Maroon" title="c4">c2</font>, <font color="Maroon" title="c5">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">  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="c3">c1</font>,<font color="Maroon" title="c2">c</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="c4">c2</font>,<font color="Maroon" title="c2">c</font><br/>  for <font color="Olive" title="b3">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</font><br/>  for <font color="Olive" title="b4">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font> &amp; <font color="Olive" title="b1">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  holds <br/><font color="Olive" title="b4">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">f1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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">f2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">c</font><br/>  for <font color="Olive" title="b2">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</font><br/>  for <font color="Olive" title="b3">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Olive" title="b1">f2</font> &amp; <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  holds <br/><font color="Olive" title="b3">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">f2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</font><br/>  for <font color="Olive" title="b2">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  holds <br/><font color="Olive" title="b2">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c8">p1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">p1</font>,<font color="Olive" title="b1">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  holds <br/><font color="Olive" title="b1">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c9">p2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">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"> (  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font> &amp; <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  implies <font color="Maroon" title="c9">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  )</span><br/>











<span class="kw">assume </span><a NAME="E1:73"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c3">c1</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c4">c2</font>,<font color="Maroon" title="c2">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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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"> (  not <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</font> or  not <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  or <font color="Maroon" title="c9">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:73"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> <a href="cat_7.html#R2" title="CAT_7:pred.2">is_pullback_of</a> <font color="Maroon" title="c6">f1</font>,<font color="Maroon" title="c7">f2</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"> (  not <font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  or <font color="Maroon" title="c9">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a>  )</span><br/>

<a NAME="E3:73"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A3</font></span>: 
( <font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">p2</font> &amp; (  for <font color="Olive" title="b1">d1</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="b2">g1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c1</font><br/>  for <font color="Olive" title="b3">g2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c2</font>  st  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c4">c2</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>  &amp; <font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> holds <br/>(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">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>  &amp;  ex <font color="Olive" title="b4">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">d</font> st <br/>( <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> &amp; (  for <font color="Olive" title="b5">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b1">d1</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g1</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b5">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g2</font> holds <br/><font color="Olive" title="b4">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">h1</font> ) ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:73"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c6">f1</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</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="c9">p2</font> is  <a href="cat_7.html#V1" title="CAT_7:attr.1">monomorphism</a> </span><br/>

<span class="kw">thus </span><a NAME="E5:73"/>
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c2</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="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_7.html#D5" target="_self" onmouseover="rs('cat_7/D5')" onmouseout="rh()">CAT_7:def 5</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="Olive" title="b1">c</font>,<font color="Maroon" title="c5">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>  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="Olive" title="b1">c</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b3">g2</font> 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="c10">d1</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="c10">d1</font>,<font color="Maroon" title="c5">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="c10">d1</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g2</font> 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="E6:73"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
 <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c5">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="c10">d1</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">g2</font> 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="c11">q1</font>, <font color="Maroon" title="c12">q2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c5">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="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c12">q2</font> implies <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E7:73"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c12">q2</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="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">p11</font> = <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">p12</font> = <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font>;<br/>
<a NAME="E8:73"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
(  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c3">c1</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>  &amp;  <a href="cat_7.html#K1" title="CAT_7:func.1">Hom</a> (<font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c4">c2</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="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="cat_7.html#T22" target="_self" onmouseover="rs('cat_7/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c7">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</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">then </span><span class="kw">consider </span><font color="Maroon" title="c15">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c5">d</font><span class="kw"> such that </span><br/><a NAME="E11:73"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
( <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c5">d</font>  st <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> &amp; <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font> holds <br/><font color="Maroon" title="c15">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:73"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="cat_7.html#D17" target="_self" onmouseover="rs('cat_7/D17')" onmouseout="rh()">Def17</a>, <a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A7</font></span></a></span>;<br/>
<a NAME="E12:73"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:73"><span class="lab"><font color="Green" title="E48">A8</font></span></a></span>;<br/>
<font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c12">q2</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c12">q2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c7">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c12">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</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="c7">f2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">p2</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" target="_self" onmouseover="rs('cat_7/T23')" onmouseout="rh()">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c6">f1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:73"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="txt" href="#E3:73"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="txt" href="#E1:73"><span class="lab"><font color="Green" title="E41">A1</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="E14:73"/>
<font color="Maroon" title="c11">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:73"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E11:73"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="txt" href="#E7:73"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E8:73"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="txt" href="#E4:73"><span class="lab"><font color="Green" title="E44">A4</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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