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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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/>







<b>thus </b><a NAME="E1:88"/>
( <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:()"><i><font color="Red">::  thesis: </font></i></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="88_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green" title="E51">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E2:88_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<b>hence </b><a NAME="E3:88_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>
 <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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">F'</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">F'</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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F'</font> being   <a href="cat_3.html#M2" title="CAT_3:mode.2">Injections_family</a> of <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">F'</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">F'</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/><b>let </b><font color="Maroon" title="c6">F'</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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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">F'</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/>



<b>assume </b><a NAME="E4:88_1"/><i><font color="Green" title="E52">A2</font></i>: 
 <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">F'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">F'</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/>

<b>reconsider </b><font color="Maroon" title="c7">oppF'</font> =  <a href="cat_3.html#K5" title="CAT_3:func.5">opp</a> <font color="Maroon" title="c6">F'</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> <b>by </b><i><a class="ref" href="cat_3.html#T75" target="_self" title="CAT_3:th.75">Th75</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">oppF'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font> )</span><br/><b>assume </b><a NAME="E1:88_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">oppF'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span><br/><b>hence </b><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>
<b>by </b><i><a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></i>
<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>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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">F'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="#E4:88_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T10" title="OPPCAT_1:th.10">OPPCAT_1:10</a></i>
<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">oppF'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></i>
<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">oppF'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="#E1:88_1_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:88_1"/><b>then </b>
 <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">oppF'</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E9:88_1"/><i><font color="Green" title="E53">A4</font></i>: 
<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>
 <b>and </b><br/><a NAME="E10:88_1"/><i><font color="Green" title="E54">A5</font></i>: 
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">oppF'</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> )
 <b>by </b><i><a class="txt" href="#E1:88_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<b>take </b>
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="E11:88_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>
 
<b>by </b><i><a class="txt" href="#E9:88_1"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>;<br/>
<b>hence </b><a NAME="E12:88_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>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T5" title="OPPCAT_1:th.5">OPPCAT_1:5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">F'</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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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/>

<b>assume </b><a NAME="E13:88_1"/><i><font color="Green" title="E55">A6</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="E14:88_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E15:88_1"/><b>then </b><i><font color="Green" title="E56">A7</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></i>;<br/>
<b>thus </b><a NAME="E16:88_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">F'</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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="88_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1_2"/><i><font color="Green" title="E57">A8</font></i>: 
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">F'</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:()"><i><font color="Red">::  thesis: </font></i></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="88_1_2_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">oppF'</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/><b>assume </b><a NAME="E1:88_1_2_1"/><i><font color="Green" title="E58">A9</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">oppF'</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="E2:88_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>
 <b>by </b><i><a class="txt" href="#E1:88_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/><a NAME="E3:88_1_2_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="txt" href="#E1:88_1_2_1"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/><a NAME="E4:88_1_2_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="ref" href="oppcat_1.html#T11" title="OPPCAT_1:th.11">OPPCAT_1:11</a></i>;<br/><a NAME="E5:88_1_2_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="txt" href="#E1:88_1_2_1"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>;<br/><a NAME="E6:88_1_2_1"/><b>then </b><i><font color="Green" title="E59">A10</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="#E13:88_1"><i><font color="Green" title="E55">A6</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E7:88_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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:88_1_2"><i><font color="Green" title="E57">A8</font></i></a>, <a class="txt" href="#E1:88_1_2_1"><i><font color="Green" title="E58">A9</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c7">oppF'</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>
<b>by </b><i><a class="txt" href="#E1:88_1_2_1"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></i>
<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>
<b>by </b><i><a class="txt" href="#E6:88_1_2_1"><i><font color="Green" title="E59">A10</font></i></a>, <a class="ref" href="oppcat_1.html#T19" title="OPPCAT_1:th.19">OPPCAT_1:19</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:88_1_2_1"><i><font color="Green" title="E58">A9</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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>
<b>by </b><i><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:88_1_2"/><b>then </b>
<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>
 
<b>by </b><i><a class="txt" href="#E10:88_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="txt" href="#E15:88_1"><i><font color="Green" title="E56">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:88_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>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E17:88_1"/><i><font color="Green" title="E57">A11</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<b>assume </b><a NAME="E18:88_1"/><i><font color="Green" title="E58">A12</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E19:88_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>
 
<b>by </b><i><a class="txt" href="#E1:88_1"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="cat_3.html#D15" target="_self" title="CAT_3:def.15">Def15</a></i>;<br/>
<a NAME="E20:88_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="#E18:88_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="cat_3.html#T45" target="_self" title="CAT_3:th.45">Th45</a></i>;<br/>
<a NAME="E21:88_1"/><b>then </b><i><font color="Green" title="E59">A13</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="#E15:88_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E22:88_1"/>
<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>
 
<b>by </b><i><a class="txt" href="#E17:88_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></i>;<br/>
<a NAME="E23:88_1"/><b>then </b>
<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">oppF'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="#E10:88_1"><i><font color="Green" title="E54">A5</font></i></a>, <a class="txt" href="#E15:88_1"><i><font color="Green" title="E56">A7</font></i></a>, <a class="txt" href="#E18:88_1"><i><font color="Green" title="E58">A12</font></i></a></i>;<br/>
<a NAME="E24:88_1"/><b>then </b>
<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">oppF'</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> 
 
<b>by </b><i><a class="txt" href="#E21:88_1"><i><font color="Green" title="E59">A13</font></i></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></i>;<br/>
<a NAME="E25:88_1"/><b>then </b>
<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">oppF'</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> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></i>;<br/>
<b>hence </b><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">oppF'</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> 
<b>by </b><i><a class="txt" href="#E18:88_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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">F'</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> 
<b>by </b><i><a class="txt" href="#E18:88_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="ref" href="cat_3.html#D6" target="_self" title="CAT_3:def.6">Def6</a></i>
<br/>.= 
<font color="Maroon" title="c6">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
<b>by </b><i><a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:88"/><i><font color="Green" title="E51">A14</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="E3:88"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></i>;<br/>
<b>hence </b><a NAME="E4:88"/>
<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>
 <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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">F'</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">F'</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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F'</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">F'</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">F'</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/><b>let </b><font color="Maroon" title="c6">F'</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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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">F'</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/>



<b>assume </b><a NAME="E5:88"/><i><font color="Green" title="E52">A15</font></i>: 
 <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">F'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">F'</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/>

<b>reconsider </b><font color="Maroon" title="c7">F'opp</font> = <font color="Maroon" title="c6">F'</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> <b>by </b><i><a class="ref" href="cat_3.html#T74" target="_self" title="CAT_3:th.74">Th74</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#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="c8">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">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="c8">x</font> )</span><br/><b>assume </b><a NAME="E1:88_2"/><i><font color="Green" title="E53">A16</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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="c8">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">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="c8">x</font></span><br/><b>hence </b><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="c8">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="c8">x</font></span>)</span>
<b>by </b><i><a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:88_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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="c8">x</font></span>)</span>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>
<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">F'</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="#E5:88"><i><font color="Green" title="E52">A15</font></i></a>, <a class="txt" href="#E1:88_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></i>
<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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">x</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:88_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="cat_3.html#D4" target="_self" title="CAT_3:def.4">Def4</a></i>
<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">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>
<b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>
<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">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>
<b>by </b><i><a class="txt" href="#E1:88_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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">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="c8">x</font>
<b>by </b><i><a class="txt" href="#E1:88_2"><i><font color="Green" title="E53">A16</font></i></a>, <a class="ref" href="cat_3.html#D3" target="_self" title="CAT_3:def.3">Def3</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E8:88"/><b>then </b>
 <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">F'opp</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#T1" target="_self" title="CAT_3:th.1">Th1</a></i>;<br/>
<b>then consider </b><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><b> such that </b><br/><a NAME="E10:88"/><i><font color="Green" title="E53">A17</font></i>: 
<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>
 <b>and </b><br/><a NAME="E11:88"/><i><font color="Green" title="E54">A18</font></i>: 
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">F'opp</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> )
 <b>by </b><i><a class="txt" href="#E2:88"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></i>;<br/>
<b>take </b>
 <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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="E12:88"/>
 <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>
 
<b>by </b><i><a class="txt" href="#E10:88"><i><font color="Green" title="E53">A17</font></i></a>, <a class="ref" href="oppcat_1.html#T14" title="OPPCAT_1:th.14">OPPCAT_1:14</a></i>;<br/>
<a NAME="E13:88"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></i>;<br/>
<b>hence </b><a NAME="E14:88"/>
 <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>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T4" title="OPPCAT_1:th.4">OPPCAT_1:4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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">F'</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/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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/>

<b>assume </b><a NAME="E15:88"/><i><font color="Green" title="E55">A19</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="E16:88"/><b>then </b><i><font color="Green" title="E56">A20</font></i>: 
<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>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T13" title="OPPCAT_1:th.13">OPPCAT_1:13</a></i>;<br/>
<b>thus </b><a NAME="E17:88"/>
( ( 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">F'</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:()"><i><font color="Red">::  thesis: </font></i></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">F'</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="88_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_3"/><i><font color="Green" title="E57">A21</font></i>: 
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">F'</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:()"><i><font color="Red">::  thesis: </font></i></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="88_3_1"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">F'opp</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/><b>assume </b><a NAME="E1:88_3_1"/><i><font color="Green" title="E58">A22</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">F'opp</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="E2:88_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>
 <b>by </b><i><a class="txt" href="#E2:88"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></i>;<br/><a NAME="E3:88_3_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="txt" href="#E1:88_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>;<br/><a NAME="E4:88_3_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="txt" href="#E1:88_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>;<br/><a NAME="E5:88_3_1"/><b>then </b>
 <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> 
 <b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>;<br/><a NAME="E6:88_3_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></i>;<br/><a NAME="E7:88_3_1"/><b>then </b><i><font color="Green" title="E59">A23</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="#E15:88"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/><a NAME="E8:88_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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 <b>by </b><i><a class="txt" href="#E1:88_3"><i><font color="Green" title="E57">A21</font></i></a>, <a class="txt" href="#E1:88_3_1"><i><font color="Green" title="E58">A22</font></i></a></i>;<br/><a NAME="E9:88_3_1"/><b>then </b>
<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">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> 
 <b>by </b><i><a class="txt" href="#E7:88_3_1"><i><font color="Green" title="E59">A23</font></i></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></i>;<br/><b>hence </b><font color="Maroon" title="c7">F'opp</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>
<b>by </b><i><a class="txt" href="#E1:88_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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>
<b>by </b><i><a class="txt" href="#E1:88_3_1"><i><font color="Green" title="E58">A22</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:88_3"/><b>then </b>
<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> 
 
<b>by </b><i><a class="txt" href="#E11:88"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E16:88"><i><font color="Green" title="E56">A20</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:88_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>
 <b>by </b><i><a class="ref" href="oppcat_1.html#T7" title="OPPCAT_1:th.7">OPPCAT_1:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E18:88"/><i><font color="Green" title="E57">A24</font></i>: 
 <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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font></span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<b>assume </b><a NAME="E19:88"/><i><font color="Green" title="E58">A25</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></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">F'</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E20:88"/>
<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>
 
<b>by </b><i><a class="txt" href="#E2:88"><i><font color="Green" title="E51">A14</font></i></a>, <a class="ref" href="cat_3.html#D18" target="_self" title="CAT_3:def.18">Def18</a></i>;<br/>
<a NAME="E21:88"/><b>then </b>
 <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> 
 
<b>by </b><i><a class="txt" href="#E19:88"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="cat_3.html#T67" target="_self" title="CAT_3:th.67">Th67</a></i>;<br/>
<a NAME="E22:88"/><b>then </b>
 <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> 
 
<b>by </b><i><a class="txt" href="#E19:88"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>;<br/>
<a NAME="E23:88"/><b>then </b>
 <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> 
 
<b>by </b><i><a class="ref" href="oppcat_1.html#T9" title="OPPCAT_1:th.9">OPPCAT_1:9</a></i>;<br/>
<a NAME="E24:88"/><b>then </b>
 <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>
 
<b>by </b><i><a class="ref" href="oppcat_1.html#D2" title="OPPCAT_1:def.2">OPPCAT_1:def 2</a></i>;<br/>
<a NAME="E25:88"/><b>then </b><i><font color="Green" title="E59">A26</font></i>: 
 <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>
 
<b>by </b><i><a class="txt" href="#E15:88"><i><font color="Green" title="E55">A19</font></i></a>, <a class="ref" href="cat_1.html#T18" title="CAT_1:th.18">CAT_1:18</a></i>;<br/>
<a NAME="E26:88"/>
<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> 
 
<b>by </b><i><a class="txt" href="#E18:88"><i><font color="Green" title="E57">A24</font></i></a>, <a class="ref" href="oppcat_1.html#T8" title="OPPCAT_1:th.8">OPPCAT_1:8</a></i>;<br/>
<a NAME="E27:88"/><b>then </b>
<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">F'opp</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">x</font>
 
<b>by </b><i><a class="txt" href="#E11:88"><i><font color="Green" title="E54">A18</font></i></a>, <a class="txt" href="#E16:88"><i><font color="Green" title="E56">A20</font></i></a>, <a class="txt" href="#E19:88"><i><font color="Green" title="E58">A25</font></i></a></i>;<br/>
<a NAME="E28:88"/><b>then </b>
<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">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> 
 
<b>by </b><i><a class="txt" href="#E19:88"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c6">F'</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>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>
<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>
<b>by </b><i><a class="txt" href="#E19:88"><i><font color="Green" title="E58">A25</font></i></a>, <a class="ref" href="cat_3.html#D5" target="_self" title="CAT_3:def.5">Def5</a></i>
<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> 
<b>by </b><i><a class="txt" href="#E25:88"><i><font color="Green" title="E59">A26</font></i></a>, <a class="ref" href="oppcat_1.html#T17" title="OPPCAT_1:th.17">OPPCAT_1:17</a></i>
<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>
<b>by </b><i><a class="ref" href="oppcat_1.html#D4" title="OPPCAT_1:def.4">OPPCAT_1:def 4</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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