<?xml version="1.0"?>
<div class="add">

<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">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Olive" title="b1">C</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Olive" title="b1">C</font> holds <br/> ( <font color="Olive" title="b2">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b3">F</font> iff <font color="Olive" title="b2">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b3">F</font> <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">c</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">C</font> holds <br/> ( <font color="Olive" title="b1">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b2">F</font> iff <font color="Olive" title="b1">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b2">F</font> <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">c</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c2">C</font> holds <br/> ( <font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Olive" title="b1">F</font> iff <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Olive" title="b1">F</font> <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="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</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"> ( <font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  )</span><br/>







<span class="kw">thus </span><a NAME="E1:93"/>
( <font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F</font> implies <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</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="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  implies <font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_1"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F</font>
 ; <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="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span><br/>

<a NAME="E2:93_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:93_1"/>
<font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">CAT_3:def 18</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">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span><br/> for <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c1">I</font>  st  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b2">F9</font> holds <br/> ex <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Olive" title="b1">d</font> holds <br/>( ( for <font color="Olive" title="b5">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b4">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b5">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">x</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>; <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">F9</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c1">I</font>  st  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Olive" title="b1">F9</font> holds <br/> ex <font color="Olive" title="b2">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> st <br/>( <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> holds <br/>( ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b3">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b4">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font> ) iff <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F9</font> be    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c5">d</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"> (  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c6">F9</font> implies  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> holds <br/>( ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:93_1"/><span class="lab"><font color="Green" title="E51">A2</font></span>: 
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c6">F9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> holds <br/>( ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b2">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">oppF9</font> =  <a href="cat_3.html#K5" title="CAT_3:func.5">opp</a> <font color="Maroon" title="c6">F9</font> as    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of  <a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T75" target="_self" title="CAT_3:th.75">Th75</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">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="c8">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"><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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">oppF9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:93_1_1"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<font color="Maroon" title="c8">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"><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="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">oppF9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span><br/><span class="kw">hence </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="c8">x</font> = 
 <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="c8">x</font></span>)</span>
<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/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <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="c8">x</font></span>)</span> <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="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <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="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_1"><span class="lab"><font color="Green" title="E52">A3</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"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c6">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93_1"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E1:93_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T10" title="OPPCAT_1:th.10">OPPCAT_1:10</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">oppF9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_1"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></span>
;<br/> <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>
<a NAME="E7:93_1"/><span class="kw">then </span>
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c7">oppF9</font>
 
<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>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font><span class="kw"> such that </span><br/><a NAME="E9:93_1"/><span class="lab"><font color="Green" title="E52">A4</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font></span>)</span>,<font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E10:93_1"/><span class="lab"><font color="Green" title="E53">A5</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font></span>)</span>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</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="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) ) )</span><br/>

<a NAME="E11:93_1"/>
<font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font></span>)</span> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:93_1"><span class="lab"><font color="Green" title="E52">A4</font></span></a>, <a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:93_1"/>
<font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T5" title="OPPCAT_1:th.5">OPPCAT_1:5</a></span>; <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">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>; <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="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</font> implies ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E13:93_1"/><span class="lab"><font color="Green" title="E54">A6</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<font color="Maroon" title="c5">d</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<a NAME="E14:93_1"/><span class="kw">then </span>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></span>;<br/>
<a NAME="E15:93_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A7</font></span>: 
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <font color="Maroon" title="c5">d</font></span>)</span>,<font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></span>;<br/>
<span class="kw">thus </span><a NAME="E16:93_1"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )
  <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="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_1_2"/><span class="lab"><font color="Green" title="E56">A8</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font>
 ; <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="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:93_1_2_1"/><span class="lab"><font color="Green" title="E57">A9</font></span>: 
<font color="Maroon" title="c10">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"> <font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span></span><br/><a NAME="E2:93_1_2_1"/>
<font color="Maroon" title="c4">F</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></span>;<br/><a NAME="E3:93_1_2_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></span>;<br/><a NAME="E4:93_1_2_1"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T11" title="OPPCAT_1:th.11">OPPCAT_1:11</a></span>;<br/><a NAME="E5:93_1_2_1"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>;<br/><a NAME="E6:93_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A10</font></span>: 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:93_1"><span class="lab"><font color="Green" title="E54">A6</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E7:93_1_2_1"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="c10">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2"><span class="lab"><font color="Green" title="E56">A8</font></span></a>, <a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> = 
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="c10">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E57">A9</font></span></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><span class="p3">(<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="c10">x</font></span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93_1_2_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a>, <a class="ref" href="oppcat_1.html#T19" title="OPPCAT_1:th.19">OPPCAT_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1_2_1"><span class="lab"><font color="Green" title="E57">A9</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="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></span>
;<br/> <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>
<a NAME="E3:93_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E15:93_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:93_1_2"/>
<font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</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">assume </span><a NAME="E17:93_1"/>
<font color="Maroon" title="c8">h</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span><br/>

<a NAME="E18:93_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E19:93_1"/><span class="lab"><font color="Green" title="E57">A12</font></span>: 
<font color="Maroon" title="c10">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"> <font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E20:93_1"/>
<font color="Maroon" title="c4">F</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_1"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></span>;<br/>
<a NAME="E21:93_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:93_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></span>;<br/>
<a NAME="E22:93_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A13</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:93_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E23:93_1"/>
<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="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93_1"><span class="lab"><font color="Green" title="E53">A5</font></span></a>, <a class="txt" href="#E15:93_1"><span class="lab"><font color="Green" title="E55">A7</font></span></a>, <a class="txt" href="#E19:93_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="txt" href="#E18:93_1"><span class="lab"><font color="Green" title="E56">A11</font></span></a></span>;<br/>
<a NAME="E24:93_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c9">k</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"><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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="#E22:93_1"><span class="lab"><font color="Green" title="E58">A13</font></span></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></span>;<br/>
<a NAME="E25:93_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c9">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="c10">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">oppF9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="#E19:93_1"><span class="lab"><font color="Green" title="E57">A12</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"><a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="#E19:93_1"><span class="lab"><font color="Green" title="E57">A12</font></span></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></span>
<br/>.= 
<font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>
;<br/> <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">assume </span><a NAME="E2:93"/><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a>  <a href="cat_3.html#R3" title="CAT_3:pred.3">is_a_coproduct_wrt</a> <font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</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="c3">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <font color="Maroon" title="c4">F</font></span><br/>

<a NAME="E3:93"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:93"/>
<font color="Maroon" title="c4">F</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">CAT_3:def 15</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">b</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c2">C</font><br/> for <font color="Olive" title="b2">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c1">I</font>  st  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b2">F9</font> holds <br/> ex <font color="Olive" title="b3">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font> st <br/>( <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b4">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b5">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b5">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b5">x</font> ) iff <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">k</font> ) ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</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">F9</font> being   <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c1">I</font>  st  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Olive" title="b1">F9</font> holds <br/> ex <font color="Olive" title="b2">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font> st <br/>( <font color="Olive" title="b2">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b4">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b4">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b4">x</font> ) iff <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k</font> ) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">F9</font> be    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c5">d</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"> (  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c6">F9</font> implies  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b3">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E5:93"/><span class="lab"><font color="Green" title="E51">A15</font></span>: 
 <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c4">F</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c6">F9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b2">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b3">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">x</font> ) iff <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k</font> ) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E52">A16</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">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="c7">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"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:93_2"/><span class="lab"><font color="Green" title="E53">A17</font></span>: 
<font color="Maroon" title="c7">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"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font></span><br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font> = 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <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="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <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="c7">x</font></span>)</span> <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="#E1:93_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>
<br/>.= 
 <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="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_3.html#K3" title="CAT_3:func.3">cods</a> <font color="Maroon" title="c6">F9</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></span>
<br/>.= 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font></span>)</span> <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="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>
<br/>.= 
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</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="c7">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E53">A17</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"><a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a> </span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_2"><span class="lab"><font color="Green" title="E53">A17</font></span></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></span>
;<br/> <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">reconsider </span><font color="Maroon" title="c7">F9opp</font> = <font color="Maroon" title="c6">F9</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  as    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c5">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></span>;<br/>
<a NAME="E8:93"/>
 <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <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="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="cat_3.html#K2" title="CAT_3:func.2">doms</a> <font color="Maroon" title="c7">F9opp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">h</font> being   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span><span class="kw"> such that </span><br/><a NAME="E10:93"/><span class="lab"><font color="Green" title="E53">A18</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E11:93"/><span class="lab"><font color="Green" title="E54">A19</font></span>: 
for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="oppcat_1.html#K2" title="OPPCAT_1:func.2">opp</a> </span>)</span>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Olive" title="b1">k</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <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="Olive" title="b2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F9opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></span>;<br/>
<span class="kw">take </span>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) ) )</span><br/>

<a NAME="E12:93"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:93"><span class="lab"><font color="Green" title="E53">A18</font></span></a>, <a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></span>;<br/>
<a NAME="E13:93"/><span class="kw">then </span>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<span class="p1">(<span class="default"><a href="oppcat_1.html#K4" title="OPPCAT_1:func.4">opp</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:93"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></span>; <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">k</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">C</font>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> holds <br/>( ( for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b2">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">x</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">k</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"> ( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">c</font> implies ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E15:93"/><span class="lab"><font color="Green" title="E55">A20</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) iff  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<a NAME="E16:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A21</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">Hom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">d</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></span>;<br/>
<span class="kw">thus </span><a NAME="E17:93"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) implies  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> implies for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:93_3"/><span class="lab"><font color="Green" title="E57">A22</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> implies <font color="Maroon" title="c7">F9opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:93_3_1"/><span class="lab"><font color="Green" title="E58">A23</font></span>: 
<font color="Maroon" title="c10">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"> <font color="Maroon" title="c7">F9opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span></span><br/><a NAME="E2:93_3_1"/>
<font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c1">I</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></span>;<br/><a NAME="E3:93_3_1"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_3_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></span>;<br/><a NAME="E4:93_3_1"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_3_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>;<br/><a NAME="E5:93_3_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>;<br/><a NAME="E6:93_3_1"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></span>;<br/><a NAME="E7:93_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A24</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:93"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/><a NAME="E8:93_3_1"/>
<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="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_3"><span class="lab"><font color="Green" title="E57">A22</font></span></a>, <a class="txt" href="#E1:93_3_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a></span>;<br/><a NAME="E9:93_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="#E7:93_3_1"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></span>;<br/><span class="kw">hence </span><font color="Maroon" title="c7">F9opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> <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="#E1:93_3_1"><span class="lab"><font color="Green" title="E58">A23</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="c9">k</font> <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"><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="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93_3_1"><span class="lab"><font color="Green" title="E58">A23</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>
;<br/> <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>
<a NAME="E3:93_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <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="#E11:93"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="txt" href="#E16:93"><span class="lab"><font color="Green" title="E56">A21</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:93_3"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</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">assume </span><a NAME="E18:93"/>
 <a href="oppcat_1.html#K6" title="OPPCAT_1:func.6">opp</a> <font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><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="Olive" title="b1">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span><br/>

<a NAME="E19:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A25</font></span>: 
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">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="c10">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"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E20:93"/><span class="lab"><font color="Green" title="E58">A26</font></span>: 
<font color="Maroon" title="c10">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"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E21:93"/>
<font color="Maroon" title="c4">F</font> <a href="cat_3.html#K4" title="CAT_3:func.4">opp</a>  is    <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> ,<font color="Maroon" title="c1">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></span>;<br/>
<a NAME="E22:93"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:93"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></span>;<br/>
<a NAME="E23:93"/><span class="kw">then </span>
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <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="c10">x</font></span>)</span> <a href="oppcat_1.html#K5" title="OPPCAT_1:func.5">opp</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:93"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>;<br/>
<a NAME="E24:93"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font> <a href="oppcat_1.html#K3" title="OPPCAT_1:func.3">opp</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></span>;<br/>
<a NAME="E25:93"/><span class="kw">then </span>
 <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></span>;<br/>
<a NAME="E26:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A27</font></span>: 
 <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</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="c10">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:93"><span class="lab"><font color="Green" title="E55">A20</font></span></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></span>;<br/>
<a NAME="E27:93"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F9opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:93"><span class="lab"><font color="Green" title="E54">A19</font></span></a>, <a class="txt" href="#E16:93"><span class="lab"><font color="Green" title="E56">A21</font></span></a>, <a class="txt" href="#E20:93"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="txt" href="#E19:93"><span class="lab"><font color="Green" title="E57">A25</font></span></a></span>;<br/>
<a NAME="E28:93"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">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="#E20:93"><span class="lab"><font color="Green" title="E58">A26</font></span></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c6">F9</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <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"><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="c10">x</font></span>)</span> <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="#E20:93"><span class="lab"><font color="Green" title="E58">A26</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"><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="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</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="#E26:93"><span class="lab"><font color="Green" title="E59">A27</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"><font color="Maroon" title="c4">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></span>
;<br/> <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>
