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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</font> being  <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">C</font><br/> for <font color="Olive" title="b3">F</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b2">f</font>,<font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Olive" title="b3">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b1">f</font>,<font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Olive" title="b2">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">I</font> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Olive" title="b1">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span><br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:91_1"/><span class="lab"><font color="Green" title="E49">A1</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span><br/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A2</font></span>:  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D17" target="_self" title="CAT_3:def.17">Def17</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c3">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="cat_3.html#T2" target="_self" title="CAT_3:th.2">Th2</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="cat_3.html#D7" target="_self" title="CAT_3:def.7">Def7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span>)</span></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:91_1"><span class="lab"><font color="Green" title="E50">A2</font></span></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="cat_3.html#D8" target="_self" title="CAT_3:def.8">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:91_1"><span class="lab"><font color="Green" title="E49">A1</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:91"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="cat_3.html#K7" title="CAT_3:func.7">*</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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